Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$\mathbf { v } = \mathbf { i } - 4 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$
A) $\mathbf { v } _ { 1 } = - \frac { 6 } { 5 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 7 } { 5 } \mathbf { i } - \frac { 14 } { 5 } \mathbf { j }$
B) $\mathbf { v } _ { 1 } = - \frac { 6 } { 5 } ( - 2 \mathrm { i } + \mathrm { j } ) , \mathbf { v } _ { 2 } = \frac { 11 } { 5 } \mathrm { i } - \frac { 14 } { 5 } \mathrm { j }$
C) $\mathbf { v } _ { 1 } = - \frac { 3 } { 2 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - 2 \mathbf { i } - \frac { 5 } { 2 } \mathbf { j }$
D) $\mathbf { v } _ { 1 } = - \frac { 6 } { 5 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = \frac { 9 } { 5 } \mathbf { i } - \frac { 22 } { 5 } \mathbf { j }$

Question

Quizplus · Accepted Answer

The vector $\mathbf{v}_1$ parallel to $\mathbf{w}$ is found using the projection formula, resulting in $\mathbf{v}_1 = -\frac{6}{5}(-2\mathbf{i} + \mathbf{j})$. The orthogonal component $\mathbf{v}_2$ is obtained by subtracting $\mathbf{v}_1$ from $\mathbf{v}$, leading to $\mathbf{v}_2 = -\frac{7}{5}\mathbf{i} - \frac{14}{5}\mathbf{j}$.

Decompose V into Two Vectors V1 and V2, Where V1 v=i−4j,w=−2i+j\mathbf { v } = \mathbf { i } - 4 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }v=i−4j,w=−2i+j

Decompose V into Two Vectors V1 and V2, Where V1 $\mathbf { v } = \mathbf { i } - 4 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$