Find the Projection of a Vector onto Another Vector
-$$\mathbf { v } = \mathbf { i } - 3 \mathbf { j } ; \mathbf { w } = 5 \mathbf { i } + 12 \mathrm { j }$$
A) $- \frac { 155 } { 169 } i - \frac { 372 } { 169 } \mathbf { j }$
B) $- \frac { 155 } { 13 } \mathbf { i } - \frac { 372 } { 13 } \mathbf { j }$
C) $- \frac { 31 } { 2 } \mathbf { i } - \frac { 186 } { 5 } \mathbf { j }$
D) $- \frac { 31 } { 10 } \mathbf { i } + \frac { 93 } { 10 } \mathbf { j }$

Question

Quizplus · Accepted Answer

The projection of $\mathbf{v}$ onto $\mathbf{w}$ is given by $\text{proj}_{\mathbf{w}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w}$. Calculating $\mathbf{v} \cdot \mathbf{w} = (1)(5) + (-3)(12) = 5 - 36 = -31$ and $\|\mathbf{w}\|^2 = 5^2 + 12^2 = 25 + 144 = 169$, the projection is $\frac{-31}{169}(5\mathbf{i} + 12\mathbf{j}) = - \frac { 155 } { 169 } \mathbf { i } - \frac { 372 } { 169 } \mathbf { j }$.

Find the Projection of a Vector onto Another Vector - v=i−3j;w=5i+12j\mathbf { v } = \mathbf { i } - 3 \mathbf { j } ; \mathbf { w } = 5 \mathbf { i } + 12 \mathrm { j }v=i−3j;w=5i+12j

Find the Projection of a Vector onto Another Vector
- $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } ; \mathbf { w } = 5 \mathbf { i } + 12 \mathrm { j }$