# Using cross product to find angle between two vectors

1. Jun 30, 2011

### yayscience

1. The problem statement, all variables and given/known data
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.

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

3. 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?

2. Jun 30, 2011

### ehild

sin(alpha)=sin(180-alpha) Plot the two vectors and you will see what angle they enclose.

ehild

3. Jun 30, 2011

### I like Serena

You might use the sign of the inner dot product to see which angle you have.

4. Jun 30, 2011

### yayscience

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.

5. Jun 30, 2011

### ehild

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

6. Jul 1, 2011

### yayscience

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