Find the component form of vector v given its magnitude is $\| \mathbf { v } \| = 7 \text { and the angle it }$ makes with the positive $x$-axis is $\theta = 120 ^ { \circ }$.
A) $\left( \frac { 7 \sqrt { 3 } } { 2 } , \frac { 7 \sqrt { 3 } } { 4 } \right)$
B) $\left( - \frac { \sqrt { 5 } } { 3 } , 7 \frac { \sqrt { 3 } } { 3 } \right)$
C) $\left( \frac { 7 \sqrt { 3 } } { 4 } , \frac { 7 \sqrt { 3 } } { 4 } \right)$
D)$\left( - \frac { 7 } { 2 } , 7 \frac { \sqrt { 3 } } { 2 } \right)$
E)$\left( \frac { 7 } { 2 } , \frac { 7 \sqrt { 5 } } { 2 } \right)$

Question

Quizplus · Accepted Answer

The component form of a vector given its magnitude $ \| \mathbf { v } \| $ and angle $ \theta $ with the positive x-axis can be found using the formulas $ v_x = \| \mathbf { v } \| \cos(\theta) $ and $ v_y = \| \mathbf { v } \| \sin(\theta) $. For $ \theta = 120^\circ $, $ \cos(120^\circ) = -\frac{1}{2} $ and $ \sin(120^\circ) = \frac{\sqrt{3}}{2} $. Therefore, $ v_x = 7 \times -\frac{1}{2} = -\frac{7}{2} $ and $ v_y = 7 \times \frac{\sqrt{3}}{2} = 7\frac{\sqrt{3}}{2} $, which matches option D.

Find the Component Form of Vector V Given Its Magnitude ∥v∥=7 and the angle it \| \mathbf { v } \| = 7 \text { and the angle it }∥v∥=7 and the angle it

Find the Component Form of Vector V Given Its Magnitude $\| \mathbf { v } \| = 7 \text { and the angle it }$