Lines in
There are several ways to write the equation of a straight line in . If the line is not parallel to the -axis, then we can write: where is the gradient of the line, and is the point where the line crosses the -axis. If we know that the line passes through the point then we can write this equation in a slightly different form:
If is another point on the line then we can calculate as .
We can also write the equation of the line in vector form: if is a vector in the direction of the line, then the line is the set of all points of the form for . Once again, if is another point on the line then we can be more precise: in this case, we can take .
Let us look again at the equation . Rearranging, we obtain: . Alternatively, . This can best be explained via a diagram:
The vector is called a normal vector to the line .
Find a normal vector to the line .
A normal vector to the line is . To see this,
Find the equation of the line with normal vector which passes through the point .
It may be useful to remember that and are orthogonal vectors.
Planes in
We saw above that in , a line can be described via an equation: , where is a normal vector to the line. What happens in ? Let’s consider a specific example: let and . Then , so the set of vectors satisfying is precisely the set of vectors of the form . This is not a line but a plane, usually called the plane.
This demonstrates the following difference between and :
- if is a non-zero vector in , then the set of vectors orthogonal to is a line,
- if is a non-zero vector in , then the set of vectors orthogonal to is a plane.
Following the same procedure as we used to find the equation of a line in , we can find the equation of a plane in .
Find the equation of the plane in with normal vector , passing through the point .