Find the projection of u onto v if , $\mathbf { u } = \langle - 5 , - 4 \rangle$ , $\mathbf { v } = \langle 2 , - 5 \rangle$ .
A) $\left\langle \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
B) $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
C) $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$
D) $\left\langle - \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$
E) $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , \frac { 20 \sqrt { 29 } } { 29 } \right\rangle$

Question

Find the projection of u onto v if , $\mathbf { u } = \langle - 5 , - 4 \rangle$ , $\mathbf { v } = \langle 2 , - 5 \rangle$ .
A)  $\left\langle \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
B)  $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 50 \sqrt { 29 } } { 29 } \right\rangle$
C)  $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$
D)  $\left\langle - \frac { 20 \sqrt { 29 } } { 29 } , - \frac { 40 \sqrt { 29 } } { 29 } \right\rangle$
E)  $\left\langle - \frac { 50 \sqrt { 29 } } { 29 } , \frac { 20 \sqrt { 29 } } { 29 } \right\rangle$

Quizplus · Accepted Answer

The projection of u onto v is given by $$	ext{proj}_v \mathbf{u} = \left(\frac{\mathbf{u}\cdot\mathbf{v}}{\Vert \mathbf{v} \Vert^2}ight)\mathbf{v}$$So, we first compute the dot product and the norm of v:$$\mathbf{u}\cdot\mathbf{v} = (-5)(2) + (-4)(-5) = 20$$$$\Vert \mathbf{v} \Vert^2 = 2^2 + (-5)^2 = 29$$Therefore, the projection of u onto v is:$$	ext{proj}_v \mathbf{u} = \left(\frac{20}{29}ight)\langle 2, -5 angle = \left\langle \frac{40}{29}, -\frac{100}{29} ightangle$$Simplifying this expression, we get:$$	ext{proj}_v \mathbf{u} = \left\langle \frac{20\sqrt{29}}{29}, -\frac{50\sqrt{29}}{29} ightangle$$Therefore, the correct answer is A.

Find the Projection of U onto V If u=⟨−5,−4⟩\mathbf { u } = \langle - 5 , - 4 \rangleu=⟨−5,−4⟩

Find the Projection of U onto V If $\mathbf { u } = \langle - 5 , - 4 \rangle$