Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 8$ and the angle it makes with the $x$-axis is $120 ^ { \circ }$.
A) $\langle - 4 \sqrt { 3 } , 4 \rangle$
B) $\langle - 8,8 \sqrt { 3 } \rangle$
C) $\langle - 8 \sqrt { 3 } , - 8 \rangle$
D) $\langle - 4,4 \sqrt { 3 } \rangle$
E) $\langle - 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$

Question

Quizplus · Accepted Answer

We use the fact that $\|\mathbf{v}\| = \sqrt{v_1^2+v_2^2}=8$. We also use the fact that the angle between $\mathbf{v}$ and the positive $x$-axis is $120^\circ$. This implies that the angle between $\mathbf{v}$ and the negative $y$-axis is $30^\circ$. Therefore, we have $$\begin{aligned}v_1 &= \|\mathbf{v}\|\cos 30^\circ = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3}\ v_2 &= -\|\mathbf{v}\|\sin 30^\circ = -8 \cdot \frac{1}{2} = -4\end{aligned}$$ Thus, the component form of $\mathbf{v}$ is $\boxed{	extbf{(D)}\; \langle-4,4\sqrt{3}angle}$.

Find the Component Form Of v\mathbf { v }v If ∥v∥=8\| \mathbf { v } \| = 8∥v∥=8

Find the Component Form Of $\mathbf { v }$ If $\| \mathbf { v } \| = 8$