Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = - 2 \mathbf { i } + 5 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$$
A) $\mathbf { v } _ { 1 } = \frac { 9 } { 5 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = \frac { 8 } { 5 } \mathbf { i } + \frac { 16 } { 5 } \mathbf { j }$
B) $\mathbf { v } _ { 1 } = \frac { 9 } { 5 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 9 } { 5 } \mathbf { i } + \frac { 26 } { 5 } \mathbf { j }$
C) $\mathbf { v } _ { 1 } = \frac { 9 } { 4 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 5 } { 2 } \mathbf { i } + \frac { 21 } { 4 } \mathbf { j }$
D) $\mathbf { v } _ { 1 } = \frac { 9 } { 5 } ( - 2 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 12 } { 5 } \mathbf { i } + \frac { 26 } { 5 } \mathbf { j }$

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Quizplus · Accepted Answer

To decompose $\mathbf{v}$ into $\mathbf{v}_1$ parallel to $\mathbf{w}$ and $\mathbf{v}_2$ orthogonal to $\mathbf{w}$, we use the projection formula for $\mathbf{v}_1 = \text{proj}_{\mathbf{w}}\mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}$. Calculating this gives $\mathbf{v}_1 = \frac{9}{5}(-2\mathbf{i} + \mathbf{j})$. Then, $\mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1$, which results in $\mathbf{v}_2 = \frac{8}{5}\mathbf{i} + \frac{16}{5}\mathbf{j}$.

Decompose V into Two Vectors V1 and V2, Where V1 v=−2i+5j,w=−2i+j\mathbf { v } = - 2 \mathbf { i } + 5 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }v=−2i+5j,w=−2i+j

Decompose V into Two Vectors V1 and V2, Where V1 $\mathbf { v } = - 2 \mathbf { i } + 5 \mathbf { j } , \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$