Question

let $|u| = 4$ at an angle of $210^{circ}$ and $|v| = 9$ at an angle of $315^{circ}$, and $w = u - v$. what is the magnitude and direction angle of $w$?
$|w| = 5.5; \theta = 156.1^{circ}$
$|w| = 5.5; \theta = 203.9^{circ}$
$|w| = 10.8; \theta = 156.1^{circ}$
$|w| = 10.8; \theta = 203.9^{circ}$

Explanation:

Step1: Find the components of \(u\) and \(v\)

For vector \(u\) with \(|u| = 4\) and \(\theta_{u}=210^{\circ}\):

• \(u_{x}=|u|\cos\theta_{u}=4\cos210^{\circ}=4\times(-\frac{\sqrt{3}}{2})=- 2\sqrt{3}\approx - 3.464\)
• \(u_{y}=|u|\sin\theta_{u}=4\sin210^{\circ}=4\times(-\frac{1}{2})=-2\)

For vector \(v\) with \(|v| = 9\) and \(\theta_{v}=315^{\circ}\):

• \(v_{x}=|v|\cos\theta_{v}=9\cos315^{\circ}=9\times\frac{\sqrt{2}}{2}\approx6.364\)
• \(v_{y}=|v|\sin\theta_{v}=9\sin315^{\circ}=9\times(-\frac{\sqrt{2}}{2})\approx - 6.364\)

Step2: Find the components of \(w = u - v\)

• \(w_{x}=u_{x}-v_{x}=-3.464 - 6.364=-9.828\)
• \(w_{y}=u_{y}-v_{y}=-2-(-6.364)=4.364\)

Step3: Calculate the magnitude of \(w\)

Use the formula \(|w|=\sqrt{w_{x}^{2}+w_{y}^{2}}\)

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Step4: Calculate the direction angle of \(w\)

Use the formula \(\theta=\arctan(\frac{w_{y}}{w_{x}})+180^{\circ}\) (since \(w_{x}<0\))

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Answer:

D. \(|w| = 10.8;\theta = 203.9^{\circ}\)