Physics I / Dot Product, Cross Product, and Applications

Practice question · Put in order

Order the steps to find the angle between u = (1,0) and v = (1,1).

Compute ‖u‖ = 1 and ‖v‖ = $\sqrt2$
$\theta = 45°$ ( $\pi/4$ radians)
Compute u · $v = 1\times1 + 0\times1 = 1$
cos $\theta$ = (u $\cdot v)$ /(‖u‖‖v‖) = 1/ $\sqrt2$

Hints

Compute the dot product and both magnitudes before anything else.
The cosine is recovered by dividing, and the angle by inverting it.

Show the answer

Compute u · $v = 1\times1 + 0\times1 = 1$
Compute ‖u‖ = 1 and ‖v‖ = $\sqrt2$
cos $\theta$ = (u $\cdot v)$ /(‖u‖‖v‖) = 1/ $\sqrt2$
$\theta = 45°$ ( $\pi/4$ radians)

Why

The dot product formula directly gives the cosine of the angle.

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