Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = \mathbf { i } + 7 \mathbf { j } , \mathbf { w } = \mathbf { i } + \mathbf { j }$$
A) $\mathbf { v } _ { 1 } = 4 ( \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - 3 \mathbf { i } + 3 \mathbf { j }$
B) $\mathbf { v } _ { 1 } = \frac { 9 } { 2 } ( \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - \frac { 7 } { 2 } \mathbf { i } + \frac { 5 } { 2 } \mathbf { j }$
C) $\mathbf { v } _ { 1 } = 4 ( \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = 3 \mathbf { i } - 3 \mathbf { j }$
D) $\mathbf { v } _ { 1 } = 8 ( \mathbf { i } + \mathbf { j } ) , \mathbf { v } _ { 2 } = - 6 \mathbf { i } + 6 \mathbf { j }$

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. The projection of $\mathbf{v}$ onto $\mathbf{w}$ gives $\mathbf{v}_1$, and $\mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1$. Calculating the projection, we find $\mathbf{v}_1 = 4(\mathbf{i} + \mathbf{j})$, and subtracting this from $\mathbf{v}$ gives $\mathbf{v}_2 = -3\mathbf{i} + 3\mathbf{j}$, matching option A.

Decompose V into Two Vectors V1 and V2, Where V1 v=i+7j,w=i+j\mathbf { v } = \mathbf { i } + 7 \mathbf { j } , \mathbf { w } = \mathbf { i } + \mathbf { j }v=i+7j,w=i+j

Decompose V into Two Vectors V1 and V2, Where V1 $\mathbf { v } = \mathbf { i } + 7 \mathbf { j } , \mathbf { w } = \mathbf { i } + \mathbf { j }$