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In the first section of this chapter we saw a couple of equations of planes. However, none of those equations had three variables in them and were really extensions of graphs that we could look at in two dimensions.

We would like a more general equation for planes. This vector is called the normal vector.

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Also notice that we put the normal vector on the plane, but there is actually no reason to expect this to be the case. We put it here to illustrate the point.

It is completely possible that the normal vector does not touch the plane in any way. Recall from the Dot Product section that two orthogonal vectors will have a dot product of zero. In other words,.

## 11.5 Lines and Planes in Space

A slightly more useful form of the equations is as follows. Start with the first form of the vector equation and write down a vector for the difference.

This second form is often how we are given equations of planes. Notice that if we are given the equation of a plane in this form we can quickly get a normal vector for the plane.

A normal vector is,. We need to find a normal vector. Recall however, that we saw how to do this in the Cross Product section.

These two vectors will lie completely in the plane since we formed them from points that were in the plane. Notice as well that there are many possible vectors to use here, we just chose two of the possibilities.

Now, we know that the cross product of two vectors will be orthogonal to both of these vectors. Since both of these are in the plane any vector that is orthogonal to both of these will also be orthogonal to the plane.

Therefore, we can use the cross product as the normal vector. This is not as difficult a problem as it may at first appear to be. We can pick off a vector that is normal to the plane. We can also get a vector that is parallel to the line.

## Chapter 2 3–Space: lines and planes

Now, if these two vectors are parallel then the line and the plane will be orthogonal. If you think about it this makes some sense. So, if the two vectors are parallel the line and plane will be orthogonal. If the line is parallel to the plane then any vector parallel to the line will be orthogonal to the normal vector of the plane.

## Chapter 2 3–Space: lines and planes

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You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. We can form the following two vectors from the given points. Show Solution This is not as difficult a problem as it may at first appear to be. So, the line and the plane are neither orthogonal nor parallel.