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Find the angle $\theta$ between the pair of vectors u and v.

$u = \begin{bmatrix} 1\\ 1 \end{bmatrix}, v = \begin{bmatrix} 7\\ 11 \end{bmatrix}$

I already figured that I'll need the lengths of the vectors so I calculated those:

||u|| = $\sqrt{2}$, and ||v|| = $\sqrt{170}$

Now what do I do?

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You will also need the dot product. –  André Nicolas Apr 16 '12 at 19:50
    
Have you not seen this formula for the connection between angles and dot products? –  The Chaz 2.0 Apr 16 '12 at 19:51
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The dot product of two unit vectors gives you the scalar projection of each onto the other. This is a signed quantity equal to the cosine of the angle between the directed vectors (when their initial points are identified): $$\frac{u}{||u||}\cdot\frac{v}{||v||}=\cos\theta$$. –  bgins Apr 16 '12 at 19:54

1 Answer 1

up vote 2 down vote accepted

If the angle is $\theta$, then $\cos \theta = \dfrac{u\cdot v}{\|u\| \times \|v\|} = (7+11)/\sqrt{2 \times 170}$. So $\theta = \cos^{-1}\dfrac{18}{\sqrt{340}}$.

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