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A line in space can be expressed using vectors. When the point and the direction vector of a line are known, the line is expressed by the vectors
where s is the name of the line, OA is the position vector of the line, a is the direction vector of the line, and n is some number.
Example 1
A line passes through space through points A (1,3,2) and B (2,4,1). Find the vector equation of the line. The direction vector of the line is the vector AB.
The vector equation of the line is
Example 2
Is the point (4,6, -1) on the line of the previous example?
A parametric representation of the line s is formed
We get a system of equations
The point is on the line only and only if the number n is found such, that the system of equations is true. Substitute the point (4,6, -1) into the system of equations.
The system of equations is true when n = 3. So the point is on the line.
A plane is formed by three points that are not on the same line. Plane direction vectors are obtained from point-to-point vectors.
Example 3
A plane passes through points A (−2,2,2), B (1,1,0) and C (2,1,2). Under what condition is point P (x, y, z) on the plane?
We define a vector representation of the plane.
Position vectors and direction vectors
The vector representation of the plane
We get a system of equations
where n and t are some numbers. The point is on the plane only and only if the numbers n and t are found such that the system of equations is true.
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