Question

Given vector u equals < -11, -5 > and vector v equals < 4, 8 >, what is proj(v,u)?

1) < 231/20, 21/4 >
2) < -231/20, -21/4 >
3) < 231/20, -21/4 >
4) < -231/20, 21/4 >

Answer 1

The projection of vector v onto vector u, calculated using the formula (v · u) / |u|^2 × u, results in < 231/20, 21/4 >, which is option 1.

Step-by-step explanation:

The student is asking for the projection of vector v onto vector u, symbolized as proj(v,u). The projection of vector v onto vector u is a vector that represents the component of v that is in the direction of u. This is calculated using the dot product of v and u divided by the magnitude of u squared, and then multiplied by vector u. The formula for this calculation is: proj(v,u) = (v · u) / |u|^2 × u, with v · u representing the dot product and |u| representing the magnitude of u.

To calculate proj(v,u), first find the dot product of vectors v and u: v · u = (4)(-11) + (8)(-5) = -44 + (-40) = -84. Next, calculate the magnitude squared of vector u: |u|^2 = (-11)^2 + (-5)^2 = 121 + 25 = 146. Now, divide the dot product by the magnitude squared of u and multiply by vector u: proj(v,u) = (-84 / 146) × <-11, -5> = < (-84 / 146)(-11), (-84 / 146)(-5) > = < 924 / 146, 420 / 146 > = < 231/20, 21/4 >.

Therefore, the correct answer is option 1) < 231/20, 21/4 >. The magnitude and direction of the projection express how much of vector v lies in the direction of vector u.