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March 11th, 2018, 09:30 PM   #1
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Math Focus: Trigonometry
[ASK] About Vector and Angle

Given w = |v|∙ u + |u| ∙ v. If θ = ∠(u ∙ w) and φ = ∠(v ∙ w) then ….
a. Φ – θ = 90°
b. θ + φ= 90°
c. θ = φ
d. θ – φ = 90°
e. θ – φ = 180°

What I have done:
$\displaystyle \cos\theta=\frac{u\cdot w}{|u||w|}$ and $\displaystyle \cos\phi=\frac{v\cdot w}{|v||w|}$
Then I substituted them as |v| and |u| to the given equation and got:
$\displaystyle w=\frac{v\cdot w}{|w|\cos\phi}\cdot u+\frac{u\cdot w}{|w|\cos\theta}\cdot v$
What to do after this? I am stuck.

Last edited by skipjack; March 12th, 2018 at 01:26 AM.
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March 12th, 2018, 07:30 AM   #2
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Let $u = |u|\hat{u}$ and $v = |v|\hat{v}$, then $w = |u||v|(\hat{u} + \hat{v})$ and $|w| = |u||v||\hat{u} + \hat{v}|$.

$\displaystyle \cos(\theta) = \frac{u\cdot w}{|u||w|} = \frac{|u|^2|v| + |u|^2|v|(\hat{u}\cdot\hat{v})}{|u|^2|v||\hat{u} + \hat{v}|}= \frac{1 + \hat{u}\cdot\hat{v}}{|\hat{u} + \hat{v}|} = \cos(\varphi)$
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