# Using cross product to find angle between two vectors

## Homework Statement

Find the angle between
\begin{align*} \vec{A} = 10\hat{y} + 2\hat{z} \\ and \\ \vec{B} = -4\hat{y}+0.5\hat{z} \end{align*}
using the cross product.

The answer is given to be 161.5 degrees.

## Homework Equations

$$\left| \vec{A} \times \vec{B} \right| = \left| \vec{A} \right| \left| \vec{B} \right|sin(\theta)$$

## The Attempt at a Solution

$$\left| \vec{A} \times \vec{B} \right| =$$ $$\left| \begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ 0 & 10 & 2 \\ 0 & -4 & 0.5 \end{array} \right| = \left| 13\hat{x} \right| = 13$$

The magnitude of A cross B is 13.

Next we find the magnitude of vectors A and B:
$$\left| \vec{A} \right| = \sqrt{10^2+2^2} = \sqrt{104} = 10.198039$$
and
$$\left| \vec{B} \right| = \sqrt{(-4)^2+(\frac{1}{2})^2} = \sqrt{16.25} = 4.0311289$$

multiplying the previous two answers we get:
41.109609

So now we should have:
$$\frac{13}{41.109609} = sin(\theta)$$

Solving for theta, we get:
18.434951 degrees.

This is frustrating: 180-18.434951 = the correct answer. I'm not quite sure where I'm going wrong here.

I must be making the same mistake repeatedly. Another problem was the same thing, but with the numbers changed, and I also got the 180-{the answer I was getting} = {the correct answer}, but when I tried the example using the SAME methodology, I got the correct answer.

Can someone please share some relevant wisdom in my direction?

Related Calculus and Beyond Homework Help News on Phys.org
ehild
Homework Helper
sin(alpha)=sin(180-alpha) Plot the two vectors and you will see what angle they enclose.

ehild

I like Serena
Homework Helper
You might use the sign of the inner dot product to see which angle you have.

I can plot them, and I can see the angle, but I'm interested in calculating the angle.
When I use the dot product I get the correct result, but I cannot see where my mistake is while using the cross product.

ehild
Homework Helper
There is no mistake, you get the sine of the angle, but there are two angles between 0 and pi with the same sine.

ehild

Oh wow; I didn't even consider that the answer wasn't unique.
Thanks!