If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 4$, and the vectors make angles of $60 ^ { \circ }$ and $80 ^ { \circ }$ with the $x$-axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$. Round answers to two decimal places.
A) $\langle 3.19,8.27 \rangle$
B) $\langle 1.81,0.39 \rangle$
C) $\langle 5.02,6.44 \rangle$
D) $\langle 3.64 , - 1.44 \rangle$
E) $\langle 2.87,8.39 \rangle$

Question

Quizplus · Accepted Answer

The correct answer is A because the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$ is $\langle 3.19,8.27 \rangle$. This can be found by using the following formulas:$$\mathbf { u } = \langle \Vert \mathbf { u } \Vert \cos \theta, \Vert \mathbf { u } \Vert \sin \theta \rangle$$$$\mathbf { v } = \langle \Vert \mathbf { v } \Vert \cos \phi, \Vert \mathbf { v } \Vert \sin \phi \rangle$$$$\mathbf { u } + \mathbf { v } = \langle \Vert \mathbf { u } \Vert \cos \theta + \Vert \mathbf { v } \Vert \cos \phi, \Vert \mathbf { u } \Vert \sin \theta + \Vert \mathbf { v } \Vert \sin \phi \rangle$$Plugging in the given values, we get:$$\mathbf { u } = \langle 5 \cos 60 ^ { \circ }, 5 \sin 60 ^ { \circ } \rangle = \langle 2.5, 4.33 \rangle$$$$\mathbf { v } = \langle 4 \cos 80 ^ { \circ }, 4 \sin 80 ^ { \circ } \rangle = \langle 0.69, 3.94 \rangle$$$$\mathbf { u } + \mathbf { v } = \langle 2.5 + 0.69, 4.33 + 3.94 \rangle = \langle 3.19, 8.27 \rangle$$

If ∥u∥=5\| \mathbf { u } \| = 5∥u∥=5 And ∥v∥=4\| \mathbf { v } \| = 4∥v∥=4

If $\| \mathbf { u } \| = 5$ And $\| \mathbf { v } \| = 4$