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March 11th, 2018, 09:30 PM  #1 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 1,998 Thanks: 132 Math Focus: Trigonometry  [ASK] About Vector and Angle
Given w = vâˆ™ u + u âˆ™ v. If Î¸ = âˆ (u âˆ™ w) and Ï† = âˆ (v âˆ™ w) then â€¦. a. Î¦ â€“ Î¸ = 90Â° b. Î¸ + Ï†= 90Â° c. Î¸ = Ï† d. Î¸ â€“ Ï† = 90Â° e. Î¸ â€“ Ï† = 180Â° What I have done: $\displaystyle \cos\theta=\frac{u\cdot w}{uw}$ and $\displaystyle \cos\phi=\frac{v\cdot w}{vw}$ 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. 
March 12th, 2018, 07:30 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,190 Thanks: 1649 
Let $u = u\hat{u}$ and $v = v\hat{v}$, then $w = uv(\hat{u} + \hat{v})$ and $w = uv\hat{u} + \hat{v}$. $\displaystyle \cos(\theta) = \frac{u\cdot w}{uw} = \frac{u^2v + u^2v(\hat{u}\cdot\hat{v})}{u^2v\hat{u} + \hat{v}}= \frac{1 + \hat{u}\cdot\hat{v}}{\hat{u} + \hat{v}} = \cos(\varphi)$ 

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angle, vector 
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