Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, $ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $.
-$$\mathbf { u } = \langle - 9 , - 4 \rangle , \mathbf { v } = \langle 7,1 \rangle$$
A) $\langle 7,1 \rangle + \langle - 1,7 \rangle$
B) $\langle - 9.38 , - 1.34 \rangle + \langle 0.38 , - 2.66 \rangle$
C) $\langle 7,1 \rangle + \langle - 16 , - 5 \rangle$
D) $\langle 3.5,0.5 \rangle + \langle - 12.5 , - 23 \rangle$

Question

Quizplus · Accepted Answer

The vector projection of $\mathbf{u}$ onto $\mathbf{v}$ is given by $\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$. Calculating this gives a vector that, when added to its orthogonal complement, equals $\mathbf{u}$. The correct orthogonal vectors are found in option B, as it correctly applies the formula for vector projection and orthogonal decomposition.

Write the Vector U as a Sum of Two Orthogonal proj⁡vu \operatorname{proj}_{\mathbf{v}} \mathbf{u} projv​u

Write the Vector U as a Sum of Two Orthogonal $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$