Vectors

The equation of a line can be represented by vectors in 3D. The line that passes through A and R can be written as:

r = a + λb

• where a is the position vector of a known point on the line

• b is the direction vector of the line (a vector parallel to the line)

• r is the position vector of any arbitrary point on the line

• λ is a scalar parameter

If you need to find the equation of a line from just two points, C and D, and you do not know the direction vector, use the fact that the vector between the two points (d - c) is the direction vector of the line:

The equation of this line is given as

r = c + λ(d-c)

Equations of 3D line in Cartesian Form

If you need to find the equation of a straight three dimensional line in Cartesian form (in terms of x, y and x), convert using the vectors a and b from the vector equation r = a + λb.

Equations of a Plane in 3D

The vector equation of a plane is given as:

r = a + λb + μc

• where a is the position vector of a known point, A, in the plane

• b and c are non-parallel, non-zero vectors in the plane

• r is the position vector of any arbitrary point, R, in the plane

• λ and μ are scalar parameters

Normal Vector

The normal vector of a plane is used to describe the direction of the plane. It is the vector that is exactly perpendicular to the plane.

Equation of a 3D plane in Cartesian Form

You can use the normal vector of a plane to write a Cartesian equation describing the plane.

Where the normal vector, n = ai + bj + ck, the Cartesian equation of the plane is given as:

ax + by + cz = d

• a, b, c and d are all constants

Scalar Product (a.k.a "Dot Product")

The scalar product, a . b, is given by the magnitude of the two vectors a and b, and by the angle between them, θ:

a . b = |a| |b| cos(θ)

• If θ = 90° (the two vectors are perpendicular), then the scalar product will equal zero.

• If a and b are parallel, a . b = |a| |b|

• If the two vectors are identical (both are a), the scalar product will equal |a|²

In Cartesian Form

Scalar Product Equation of a Plane

The equation of a plane can also be written as the scalar product of the normal vector to the plane and the position vector of any arbitrary point on the plane:

r . n = k

• r is the position vector of any arbitrary point on the plane

• n is the normal vector of the plane

• k is a scalar constant for the plane, where k = a . n for a specific point in the plane with position vector a