Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } , \mathbf { w } = 3 \mathbf { i } + \mathbf { j }$$
A) $\mathbf { v } _ { 1 } = - \frac { 3 } { 10 } ( 3 \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 11 } { 10 } \mathbf { i } + \frac { 33 } { 10 } \mathbf { j }$
B) $\mathbf { v } _ { 1 } = - \frac { 3 } { 10 } ( 3 i + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 17 } { 10 } \mathbf { i } + \frac { 12 } { 5 } \mathbf { j }$
C) $\mathbf { v } _ { 1 } = - \frac { 1 } { 3 } ( 3 i + j ) , v _ { 2 } = - 1 i + \frac { 10 } { 3 } j$
D) $\mathbf { v } _ { 1 } = - \frac { 3 } { 10 } ( 3 i + j ) , v _ { 2 } = \frac { 7 } { 10 } \mathbf { i } + \frac { 39 } { 10 } \mathbf { j }$

Question

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 $\mathbf{v}_1 = -\frac{3}{10}(3\mathbf{i} + \mathbf{j})$. Then, $\mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1$, which results in $\mathbf{v}_2 = -\frac{11}{10}\mathbf{i} + \frac{33}{10}\mathbf{j}$.

Decompose V into Two Vectors V1 and V2, Where V1 v=−2i+3j,w=3i+j\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } , \mathbf { w } = 3 \mathbf { i } + \mathbf { j }v=−2i+3j,w=3i+j

Decompose V into Two Vectors V1 and V2, Where V1 $\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } , \mathbf { w } = 3 \mathbf { i } + \mathbf { j }$