Given the vector $$ \mathbf{u} $$ equal to $$ 2\left\langle\cos 115^{\circ}, \sin 115^{\circ} ight angle $$ and vector $$ \mathbf{v} $$ equal to $$ 6\left\langle\cos 325^{\circ}, \sin 325^{\circ} ight angle $$, find the sum $$ \mathbf{u}+\mathbf{v} $$ and write your answer in magnitude and direction form with the magnitude rounded to the nearest tenth and the direction rounded to the nearest degree, $$ 0^{\circ} \leq heta

Question

Given the vector $$ \mathbf{u} $$ equal to $$ 2\left\langle\cos 115^{\circ}, \sin 115^{\circ}ightangle $$ and vector $$ \mathbf{v} $$ equal to $$ 6\left\langle\cos 325^{\circ}, \sin 325^{\circ}ightangle $$, find the sum $$ \mathbf{u}+\mathbf{v} $$ and write your answer in magnitude and direction form with the magnitude rounded to the nearest tenth and the direction rounded to the nearest degree, $$ 0^{\circ} \leq 	heta<360^{\circ} $$.

UpStudy AI · Accepted Answer

To find the sum of the vectors $$ \mathbf{u} $$ and $$ \mathbf{v} $$ and express it in magnitude and direction form, follow these steps:

1. **Express Each Vector in Cartesian Coordinates:**
   
   Given:
   $$
   \mathbf{u} = 2\langle \cos 115^\circ, \sin 115^\circ \rangle
   $$
   $$
   \mathbf{v} = 6\langle \cos 325^\circ, \sin 325^\circ \rangle
   $$
   
   Calculate the components:
   $$
   \cos 115^\circ \approx -0.4226, \quad \sin 115^\circ \approx 0.9063
   $$
   $$
   \cos 325^\circ \approx 0.8192, \quad \sin 325^\circ \approx -0.5736
   $$
   
   Therefore:
   $$
   \mathbf{u} = 2 \times \langle -0.4226, 0.9063 \rangle = \langle -0.8452, 1.8126 \rangle
   $$
   $$
   \mathbf{v} = 6 \times \langle 0.8192, -0.5736 \rangle = \langle 4.9152, -3.4416 \rangle
   $$

2. **Add the Vectors Component-Wise:**
   $$
   \mathbf{u} + \mathbf{v} = \langle -0.8452 + 4.9152, 1.8126 - 3.4416 \rangle = \langle 4.07, -1.63 \rangle
   $$

3. **Calculate the Magnitude of the Sum:**
   $$
   \text{Magnitude} = \sqrt{4.07^2 + (-1.63)^2} \approx \sqrt{16.56 + 2.66} \approx \sqrt{19.22} \approx 4.4
   $$

4. **Determine the Direction of the Sum:**
   $$
   \theta = \arctan\left(\frac{-1.63}{4.07}\right) \approx \arctan(-0.400) \approx -21.8^\circ
   $$
   Since the vector is in the fourth quadrant (positive $$ x $$, negative $$ y $$):
   $$
   \theta = 360^\circ - 21.8^\circ \approx 338^\circ
   $$

**Final Answer:**
The sum $$ \mathbf{u} + \mathbf{v} $$ has a magnitude of 4.4 and direction 338°.
The sum of vectors $$ \mathbf{u} $$ and $$ \mathbf{v} $$ is a vector with a magnitude of 4.4 and a direction of 338 degrees.

Given the vector equal to and vector equal to
, find the sum and write your answer in magnitude and direction
form with the magnitude rounded to the nearest tenth and the direction rounded to the nearest
degree, .

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