Question

How do you find the angle between the vectors `v=i+j, w=3i-j`?

Answer 1

Compute `v*w`
Compute `|v| and |w|`
The angle between `theta = cos^-1((v*w)/(|v||w|))`

Explanation:

The dot product of two vectors of the form `a = a_xhati + a_yhatj and a = b_xhati + b_yhatj` is:

`a*b = a_x(b_x) + a_y(b_y)" [1]"`

Use the pattern of equation [1] to compute the dot product of the two given vectors:

`v*w = 1(3) + 1(-1) = 2" [2]"`

To compute the dot product of two vectors in polar form, one would use formula:

`v*w = |v||w|cos(theta)" [3]"`

where `theta` is the angle between the two vectors.

Compute the magnitudes of the two vectors

`|v| = sqrt(1^2 + 1^2) = sqrt(2)`

`|w| = sqrt(3^2 + (-1)^2) = sqrt(10)`

Solve equation [3] for `theta` and substitute the known values:

`theta = cos^-1((v*w)/(|v||w|))`

`theta = cos^-1(2/(sqrt2sqrt10))`

`theta ~~ 63.4^@`