WEBVTT
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Angle between Two Vectors in Space
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In this video, weโre going to learn how we can find the angle between any two vectors in space by using the dot product.
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And weโll see how to do this in a few situations, for example, given the component forms of a vector or given a graphical representation of the vectors.
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To do this, weโre first going to need to recall a couple of facts about vectors.
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First, we know how to find the dot product of two vectors of equal dimensions.
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If ๐ฎ is the vector with components ๐ฎ one, ๐ฎ two, up to ๐ฎ ๐ and ๐ฏ is the vector with components ๐ฏ one, ๐ฏ two, up to ๐ฏ ๐, then the dot product between ๐ฎ and ๐ฏ is equal to the sum of the products of the corresponding components.
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๐ฎ dot ๐ฏ is ๐ฎ one ๐ฏ one plus ๐ฎ two ๐ฏ two and we sum all the way up to ๐ฎ ๐ times ๐ฏ ๐.
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And weโve seen a few different ways we can apply the dot product.
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For example, if ๐ is the angle between vectors ๐ฎ and ๐ฏ, then we know the cos of ๐ will be equal to the dot product between ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ.
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And itโs worth pointing out thereโs a second way of viewing this formula.
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If we let ๐ฎ hat be the unit vector pointing in the same direction as vector ๐ฎ and ๐ฏ hat be the unit vector pointing in the same direction as vector ๐ฏ โ so ๐ฎ hat is ๐ฎ divided by the magnitude of ๐ฎ and ๐ฏ hat is ๐ฏ divided by the magnitude of ๐ฏ โ then the cos of ๐ will also be equal to the dot product between ๐ฎ hat and ๐ฏ hat.
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This gives us a nice geometric interpretation of the dot product.
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This formula is what weโre going to use to find the angle between our two vectors.
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Weโll calculate this expression and then take the inverse cos of both sides of the equation.
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However, there is one thing worth pointing out here about our value of ๐.
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We recall if weโre working in degrees, then the inverse cosine function will have a range between zero and 180 degrees inclusive.
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So if we only take the inverse cos of this expression, our answer will always be between zero and 180 degrees inclusive.
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And this has a useful result geometrically.
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If we sketch the vectors ๐ฎ and ๐ฏ starting at the same point, then by using this formula to find the value of ๐, we will always get the smaller angle.
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And of course, we can find the other angle directly from the sketch.
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These two angles sum to give us 360.
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So its angle will be 360 degrees minus ๐.
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And an alternative method of seeing why this is true is to think about what happens when we take the inverse cos of both sides of the equation.
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We know thereโs multiple solutions for this.
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And we also know that if ๐ is a solution to this, then 360 minus ๐ is also a solution because the cos of ๐ is equal to the cos of 360 minus ๐.
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The last thing weโre going to point out is that all of what we have just discussed is true if instead we were working in radians.
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However, our values of ๐ would range between zero and ๐ instead.
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Letโs now see some examples of how weโre going to apply this to find the angle between two vectors.
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Given that the modulus of vector ๐ is 35 and the modulus of vector ๐ is 23 and the dot product between ๐ and ๐ is equal to negative 805 root two divided by two, determine the measure of the smaller angle between the two vectors.
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In this question, weโre given some information about vectors ๐ and ๐.
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And weโre asked to determine the smaller angle between these two vectors.
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Sometimes in these questions we like to sketch a picture of whatโs happening.
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However, the information weโre given about our vectors wonโt allow us to sketch a picture.
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We donโt know the components of vectors ๐ and ๐.
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Instead, we only know their modulus and their dot product.
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So weโre going to need to rely entirely on our formula.
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Remember, this tells us if ๐ is the angle between two vectors ๐ and ๐, then the cos of ๐ will be equal to the dot product between ๐ and ๐ divided by the modulus of ๐ times the modulus of ๐.
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And we would find the value of ๐ by taking the inverse cosine of both sides of this equation.
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And this gives us a useful result because the inverse cosine function has a range between zero and 180 degrees.
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Therefore, it doesnโt really matter how we draw our vectors ๐ and ๐.
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If the value of ๐ is between zero and 180, it will always give us the smaller angle between these two vectors.
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The only possible caveat to this would be if of our vectors point in exactly opposite directions.
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Then the angle measured in both directions will be equal to 180 degrees.
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However, as weโll see, thatโs not whatโs happening in this question.
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Letโs now find the smaller angle between our two vectors ๐ and ๐.
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It solves the equation the cos of ๐ will be equal to the dot product between ๐ and ๐ divided by the modulus of ๐ times the modulus of ๐.
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In the question, weโre told the dot product between ๐ and ๐ is equal to negative 805 root two over two, the modulus of ๐ is equal to 35, and the modulus of ๐ is equal to 23.
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So we can substitute these values directly into our formula, giving us the cos of ๐ is negative 805 root two over two all divided by 35 times 23.
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We can simplify this.
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Remember, dividing by a number is the same as multiplying by the reciprocal of that number, giving us the cos of ๐ is negative 805 root two divided by two times 35 times 23.
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And if we were to evaluate 35 times 23, we would see itโs exactly equal to 805.
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So we can cancel these, leaving us with the cos of ๐ is equal to negative root two over two.
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And finally, we can solve for our value of ๐ by taking the inverse cos of both sides of the equation.
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Remember, we know this will give us the smaller angle between our two vectors.
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This gives us ๐ is the inverse cos of negative root two over two, which we can calculate is 135 degrees.
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Letโs now see an example of how we would calculate the angle between two vectors given their component forms.
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Find the angle ๐ between the vectors ๐ four, two, negative one and ๐ eight, four, negative two.
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In this question, weโre given two vectors in component form.
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And weโre asked to find the angle ๐ between these two vectors.
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To do this, we know a formula for finding the angle between any two vectors.
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We recall if ๐ is the angle between vectors ๐ and ๐, then ๐ satisfies the equation the cos of ๐ is equal to the dot product between ๐ and ๐ divided by the modulus of ๐ times the modulus of ๐.
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And since weโre given ๐ and ๐ in component form, we can calculate all of these values.
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So we can find our value of ๐.
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Letโs start by calculating the dot product between ๐ and ๐.
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So we want to find the dot product between the vectors four, two, negative one and eight, four, negative two.
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Remember, to find a dot product between two vectors, we need to multiply the corresponding components together and then add all of these together.
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Multiplying the first components of each vector together, we get four times eight.
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Multiplying the second components, we get two times four.
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And multiplying the third components, we get negative one times negative two.
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So the dot product of these two vectors will be the sum of these three products.
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And we can evaluate this expression.
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We get the dot product between ๐ and ๐ is 42.
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However, this is not the only thing we need to calculate.
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We also need to find the modulus of ๐ and the modulus of ๐.
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To do this, weโre first going to need to recall how we find the modulus of a vector.
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Remember, the modulus of a vector is the square root of the sum of the squares of its components.
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In other words, the modulus of the vector ๐, ๐, ๐ is the square root of ๐ squared plus ๐ squared plus ๐ squared.
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And we know the components of ๐ are four, two, and negative one.
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So the modulus of ๐ is the square root of four squared plus two squared plus negative one squared.
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And if we evaluate this expression, we see itโs equal to the square root of 21.
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We can then do exactly the same thing to find the modulus of ๐.
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Itโs equal to the square root of eight squared plus four squared plus negative two all squared.
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And if we were to evaluate and simplify this expression, we would see that the modulus of ๐ is two root 21.
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Now that weโve found the dot product between ๐ and ๐ and the modulus of ๐ and the modulus of ๐, we can substitute these into our equation involving ๐.
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We showed the dot product between ๐ and ๐ is 42, the modulus of ๐ is root 21, and the modulus of ๐ is two root 21.
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Therefore, the cos of ๐ is 42 over root 21 times two root 21.
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However, if we start evaluating this expression, we see something interesting.
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In the denominator of this expression, root 21 multiplied by two root 21 simplifies to give us 42.
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And 42 over 42 is one.
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So our entire equation simplifies to give us the cos of ๐ is equal to one, and we can solve for ๐.
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We take the inverse cosine of both sides of the equation, giving us ๐ is the inverse cos of one, which we know is zero degrees.
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And we could stop here; however, this gives us a useful piece of information.
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If the angle between ๐ and ๐ is zero degrees, then ๐ and ๐ point in the same direction.
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In other words, weโve also shown that the vectors ๐ and ๐ are parallel.
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And in fact, we could directly prove this.
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We would see that our vector ๐ is just two times the vector ๐.
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And thereโs a useful result we can get from this.
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Because our scalar is positive, the angle between these two vectors will be zero.
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However, if this scalar was negative, then the angle between them would be 180 degrees because then our vectors would point in exactly opposite directions.
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Either way, we were able to show the angle ๐ between the vectors ๐ and ๐ given to us in the question was zero degrees.
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Letโs see another example of finding the angle between two vectors.
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Find the angle ๐ between the vectors ๐ฏ is equal to ๐ข and ๐ฐ is equal to three ๐ข plus two ๐ฃ plus four ๐ค.
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Give your answer correct to two decimal places.
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In this question, weโre given two vectors ๐ฏ and ๐ฐ in terms of the unit directional vectors ๐ข, ๐ฃ, and ๐ค.
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And weโre asked to find the angle ๐ between these two vectors.
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And we need to give our answer to two decimal places.
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To do this, we can start by recalling we have a formula to find the angle between two vectors.
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Since ๐ is the angle between vectors ๐ฏ and ๐ฐ, the cos of ๐ must be equal to the dot product between ๐ฏ and ๐ฐ divided by the magnitude of ๐ฏ times the magnitude of ๐ฐ.
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So to find the value of ๐, we need to find the dot product between ๐ฏ and ๐ฐ, the magnitude of ๐ฏ, and the magnitude of ๐ฐ.
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Then all weโll need to do is take the inverse cosine of both sides of the equation.
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Thereโs several different ways of doing this.
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For example, we could work directly with the unit directional vector notation for ๐ฏ and ๐ฐ.
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However, we could also write these vectors component-wise by taking the coefficients of the unit directional vectors.
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Either method will work; itโs personal preference which weโll use.
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Weโll write ๐ฏ and ๐ฐ component-wise.
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๐ฏ is the vector one, zero, zero and ๐ฐ is the vector three, two, four.
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Letโs now start finding the values of our equation.
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Letโs start with the dot product between ๐ฏ and ๐ฐ.
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Remember, to find the dot product of two vectors, we need to find the product of the corresponding components and then add the results together.
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In this case, the first component of ๐ฏ times the first component of ๐ฐ is one times three, the second component of ๐ฏ times the second component of ๐ฐ is zero times two, and the third component of ๐ฏ times the third component of ๐ฐ is zero times four.
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So the dot product is the sum of these, one times two plus zero times two plus zero times four.
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And we can just calculate this expression.
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The second and third terms are zero.
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So this just gives us three.
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We now want to find the magnitude of our two vectors.
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Letโs start with the magnitude of ๐ฏ.
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Remember, ๐ฏ is the unit directional vector ๐ข.
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And remember, ๐ข is a unit directional vector.
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It has magnitude one.
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Now to find the magnitude of ๐ฐ is more difficult.
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So weโll write this in component notation.
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And weโll recall to find the magnitude of a vector, we need to find the square root of the sum of the squares of the components.
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So the magnitude of the vector ๐, ๐, ๐ will be the square root of ๐ squared plus ๐ squared plus ๐ squared.
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So for vector ๐ฐ, this will be the square root of three squared plus two squared plus four squared, which we can calculate is equal to the square root of 29.
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Now that we found these values, we can substitute them into our equation for ๐.
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We showed the dot product between ๐ฏ and ๐ฐ is three, the magnitude of ๐ฏ is one, and the magnitude of ๐ฐ is root 29.
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So we must have the cos of ๐ is three divided by one times the square root of 29.
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And now we can solve the ๐ by taking the inverse cos of both sides of our equation.
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This gives us that ๐ is the inverse cos of three divided by root 29.
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And if we calculate this and round our answer to two decimal places, we see that ๐ is 56.15 degrees.
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Letโs now see an example of how we would find the angle between two vectors given in a diagram.
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Find the angle between the vectors shown in the following diagram.
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In this question, we need to find the angle between two vectors.
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And weโre given these vectors on a diagram.
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And we can also see the angle between the two vectors given on our diagram.
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We have a few different options of how we could calculate this.
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For example, we could just do this by using trigonometry.
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However, we also have a formula for finding the angle between two vectors.
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We recall if ๐ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ will be equal to the dot product of ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ.
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We can then use this to solve for the value of ๐ by taking the inverse cosine of both sides of the equation.
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So to answer this question, weโre going to need to find the dot product between our vectors ๐ฎ and ๐ฏ and the magnitude of ๐ฎ and the magnitude of ๐ฏ.
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And to do this, weโre going to want to write our vectors in component form.
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Weโll do this by using the diagram.
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Letโs start with our vector ๐ฎ.
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We could see on our diagram it starts at the origin and then at the endpoint it has an ๐ฅ-coordinate of negative two root three.
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So the change in ๐ฅ is negative two root three.
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Similarly, we can see its ๐ฆ-coordinate starts at zero and ends at two and its change in ๐ฆ is two.
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So we can represent ๐ฎ as the vector with horizontal component negative two root three and vertical component two.
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We can do exactly the same for vector ๐ฏ.
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We can see it starts at the origin and then ends at an ๐ฅ-coordinate of negative two and it starts at the origin and ends at a ๐ฆ-coordinate of negative two.
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So the change in ๐ฅ is negative two, and the change in y is negative two.
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So ๐ฏ is the vector negative two, negative two.
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Now we need to find the dot product of these two vectors and their magnitudes.
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Letโs start by calculating the dot product between ๐ฎ and ๐ฏ.
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Remember, to find the dot product of two vectors, we need to find the products of corresponding components and then add all of these together.
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So we multiply the first components of ๐ฎ and ๐ฏ together to give us negative two root three times negative two.
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And then we add the products of their second components.
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Thatโs two multiplied by negative two.
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And if we calculate this expression, we see itโs equal to four root three minus four.
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But weโre not done yet.
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We still need to find the magnitude of ๐ฎ and the magnitude of ๐ฏ.
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Letโs start by finding the magnitude of ๐ฎ.
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Remember, we can find this by taking the square root of the sums of the squares of the components.
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So the magnitude of ๐ฎ is the square root of negative two root three all squared plus two squared, which, we can calculate, gives us the square root of 12 plus four, which is root 16, which is of course just equal to four.
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We can then do exactly the same to find the magnitude of ๐ฏ.
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We square each component of ๐ฏ, add these together, and take the square root.
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The magnitude of ๐ฏ is the square root of negative two squared plus negative two squared, which of course simplifies to give us the square root of four plus four, which is equal to root eight.
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And now that we found all of these values, weโre ready to substitute them into our equation for ๐.
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Substituting in ๐ฎ dot ๐ฏ is four root three minus four, the magnitude of ๐ฎ is four, and the magnitude of ๐ฏ is root eight, we get the cos of ๐ is four root three minus four all divided by four root eight.
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And itโs worth pointing out here we can simplify this expression to give us root six minus root two all divided by four.
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However, itโs not necessary because all we need to do now is take the inverse cosine of both sides of the equation.
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This gives us that ๐ is the inverse cos of root six minus root two all divided by four, which, we can calculate, gives us 75 degrees.
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And this is our final answer because if we look in our diagram, there are two possible angles between vectors ๐ฏ and ๐ฎ.
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Thereโs the angle shown in our diagram and thereโs the angle we could take in the opposite direction.
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However, this secondary angle shown in green is bigger than 75 degrees, so it canโt possibly be 75 degrees.
00:14:52.520 --> 00:14:56.080
In fact, it would be 360 minus 75 degrees.
00:14:56.240 --> 00:15:02.530
Therefore, we were able to show the angle between the two vectors in our diagram ๐ฎ and ๐ฏ is given by 75 degrees.
00:15:03.380 --> 00:15:08.370
Letโs now go through one last example of how we can use our formula to find information about vectors.
00:15:10.740 --> 00:15:14.400
The angle between vector ๐ and vector ๐ is 22 degrees.
00:15:14.430 --> 00:15:23.080
If the magnitude of vector ๐ is equal to three times the magnitude of vector ๐ is equal to 25.2, find the dot product between ๐ and ๐ to the nearest hundredth.
00:15:23.530 --> 00:15:26.850
In this question, weโre given some information about two vectors ๐ and ๐.
00:15:26.890 --> 00:15:30.730
First, weโre told the angle between these two vectors is equal to 22 degrees.
00:15:30.970 --> 00:15:33.710
Next, weโre also told information about their magnitudes.
00:15:33.710 --> 00:15:40.690
We know the magnitude of ๐ is equal to 25.2, and we know that three times the magnitude of ๐ is also equal to 25.2.
00:15:40.860 --> 00:15:44.260
So the magnitude of ๐ is three times bigger than the magnitude of ๐.
00:15:44.470 --> 00:15:47.220
We need to use this to find the dot product of ๐ and ๐.
00:15:47.250 --> 00:15:49.560
And we need to give our answer to the nearest hundredth.
00:15:49.880 --> 00:15:56.310
To answer this question, we need to notice that we know a formula which connects the angle between two vectors with their dot product.
00:15:56.470 --> 00:16:07.120
We recall if ๐ is the angle between two vectors ๐ and ๐, then we know that the cos of ๐ must be equal to the dot product between ๐ and ๐ divided by the magnitude of ๐ times the magnitude of ๐.
00:16:07.320 --> 00:16:09.620
And in this question, we already know some of these values.
00:16:09.650 --> 00:16:13.520
For example, weโre told the angle between our two vectors is 22 degrees.
00:16:13.670 --> 00:16:18.020
Next, weโre also told the magnitude of ๐ is equal to 25.2.
00:16:18.260 --> 00:16:22.380
And we could also find the magnitude of ๐ using the information given to us in the question.
00:16:22.530 --> 00:16:27.630
One way of doing this is to notice that three times the magnitude of ๐ is equal to 25.2.
00:16:27.790 --> 00:16:32.820
We can then solve this to find the magnitude of ๐ by dividing both sides of our equation through by three.
00:16:33.240 --> 00:16:36.570
And calculating this, we get the magnitude of ๐ is 8.4.
00:16:36.720 --> 00:16:40.620
So, in fact, the only unknown in this equation is the dot product between ๐ and ๐.
00:16:40.620 --> 00:16:42.430
And thatโs exactly what weโre asked to calculate.
00:16:42.650 --> 00:16:51.620
So weโll substitute the angle of ๐ equal to 22 degrees, the magnitude of ๐ equal to 25.2, and the magnitude of ๐ equal to 8.4 into our equation.
00:16:51.870 --> 00:16:59.420
This gives us the cos of 22 degrees should be equal to the dot product between ๐ and ๐ divided by 25.2 times 8.4.
00:16:59.700 --> 00:17:03.440
And now we can just rearrange this equation for the dot product between ๐ and ๐.
00:17:03.580 --> 00:17:06.640
We multiply through by 25.2 times 8.4.
00:17:06.850 --> 00:17:12.170
This gives us ๐ dot ๐ is 25.2 times 8.4 times the cos of 22 degrees.
00:17:12.340 --> 00:17:15.770
And we can calculate this to the nearest hundredth or to two decimal places.
00:17:15.800 --> 00:17:18.260
Itโs equal to 196.27.
00:17:20.620 --> 00:17:22.420
Letโs now go over the key points of this video.
00:17:22.590 --> 00:17:31.570
First, we know if ๐ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ will be equal to the dot product of ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ.
00:17:31.790 --> 00:17:34.750
And this works so long as neither vector ๐ฎ or ๐ฏ is equal to zero.
00:17:34.900 --> 00:17:38.310
And to take the dot product of ๐ฎ and ๐ฏ, we need them to have the same dimension.
00:17:38.520 --> 00:17:43.990
We can also use this formula to find the angle between two vectors by taking the inverse cosine of both sides of this equation.
00:17:44.180 --> 00:17:53.470
But we do need to be careful because the inverse cosine function has a range between zero and 180 degrees inclusive, or if weโre working in radians, zero to ๐ inclusive.
00:17:53.810 --> 00:17:57.730
So this method will give us the smaller of the two angles between our vectors ๐ฎ and ๐ฏ.