# What's the acute angle between the planes to the nearest thousandth of a radian?

## $8 x + 4 y + 3 z = 1$ and $7 x + 10 y + 7 z = - 6$ Solving the angle between vectors is simple, but I do not know how to go about finding them between planes, is there a formula that can be used?

May 30, 2018

See below

#### Explanation:

The angle between two planes is the angle between their normal vector.

When a plane is given in the $a x + b y + c z + d = 0$ form, then it's normal vector has coordinates $\left(a , b , c\right)$.

So, if we call ${\Pi}_{1}$ the first plane and ${\Pi}_{2}$ the second, we have

${n}_{1} = \left(8 , 4 , 3\right)$ (normal vector to ${\Pi}_{1}$)
${n}_{2} = \left(7 , 10 , 7\right)$ (normal vector to ${\Pi}_{2}$)

From here, you only need to find the angle between two vectors, which you said you're familiar with.

Let me know if you need help with the last step as well!