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

Question

Quizplus · Accepted Answer

We can start by sketching the vector in the coordinate plane. Since it makes an angle of $120^\circ$ with the $x$-axis, it will lie in the second quadrant, and will form a $30^\circ$ angle with the $y$-axis.

[asy]
size(6cm);
import olympiad;
import markers;

draw((-1,0)--(2.8,0),Arrow);
draw((0,-1)--(0,2.8),Arrow);

draw((0,0)--dir(120),red,Arrow);

draw(rightanglemark((0,0),dir(120),(0,1)));
draw((0,0.2)--(0.2,0.2)--(0.2,0));

label("$	heta$", (0.3,0.3), NE, red);
label("$x$", (2.8,0), E);
label("$y$", (0,2.8), N);

label("$\mathbf{v}$", dir(120), W, red);
[/asy]

We can see that the $y$-component of $\mathbf{v}$ is $\|\mathbf{v}\|\sin 30^\circ=2$, and that the $x$-component of $\mathbf{v}$ is $\|\mathbf{v}\|\cos 30^\circ=2\sqrt{3}$. Therefore, the component form of $\mathbf{v}$ is $\boxed{\langle -2,2\sqrt{3}angle}$.

Find the Component Form Of v\mathbf { v }v If ∥v∥=4\| \mathbf { v } \| = 4∥v∥=4

Find the Component Form Of $\mathbf { v }$ If $\| \mathbf { v } \| = 4$