# Distance between Two Points

# Distance between a Point and a Line

So, you have your point P, and all you need to do is pick an arbitrary point on the line, Q. You also have the vector part of your line, $< a,b,c>$. An important thing to remember is the

**cross product**of two vectors equals

**the area of the parallelogram**made by those two vectors. The area of a parallelogram also equals

**the base $\cdot$ height**.

Notice that the area equals $< a,b,c> * $ the distance to the line. So, we could write an equation for the distance from a point to a line. $$\dfrac{|\vec{PQ}\times{}< a,b,c>|}{|< a,b,c>|}$$ The numerator is the area of the parallelogram, and the denominator is the base. So, the quotient is the distance to the line.

## Example

# Distance between a Point and a Plane

This is a little bit different, here we have to use something called a **projection**, where one vector "casts a shadow" onto another vector.

## Example

# Distance between Two Skew Lines

**cross product**. Now, we have a normal vector $< A,B,C>$ and we must pick a point on one of the lines to define our plane. Last, we pick a point on the other line, and take the distance between the plane and that point.

# Distance between Two Parallel Planes

First, you **must **verify the planes are parallel. The way to do this is to check if the normal vectors are the same or are multiples of each other. Then, you pick a point on one plane, and find the distance between that point and the other plane.

David Witten