In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv

Video transcript

Hello, today we're going to be calculating the following projection. So we are asked to calculate the projection of A onto B. Now, the projection of A on to B is defined as the dot product of A and B divided by the magnitude of B squared. And that value will be multiplied by the vector B. So the first thing we'll need to do is we need to calculate the dot product of A and B. Then we'll need to calculate the magnitude of B squared vector A is given to us as six I plus J and vector B is given to us as I minus 27 J. In order to calculate the dot product of A and B, we'll need to multiply both of the I components together multiply the J components together. Then we'll take the sum of both of those products. So the dot product of A and B is equal to six multiplied by one plus one multiplied by negative 27 six, multiplied by one will give us the value of six and one multiplied by negative 27 will give us the value of negative 27 and six minus 27 will give us a final value of negative 21. So that means the dot product of A and B is equal to negative 21. Next, let's calculate the magnitude of B squared. Now the magnitude of B squared is just the sum of the components of vector B. So the magnitude of B squared is equal to one squared plus negative 27 squared. One squared will give us the value of one and negative 27 squared will give us the value of positive and one plus 729 is equal to 730. Now that we have the dot product of A and B and the magnitude of B squared, we can use these values to calculate the projection of A onto B. Now, the projection of A on to B is defined as the top product of A and B which is negative 21 over the magnitude of B squared, which is 730 multiplied by vector B which was given to us as I minus 27 J, negative 21/730 is in its simplest form. So we'll need to distribute negative 21/730 into vector B. And that will give us a final projection of negative 21/730 I plus 567/ J. This will be the value of the projection of A onto B. And with that being said, the answer to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.