Equations of lines in 3D (HL)

Direction vectors are analogous to gradients.

In vector geometry, a line is identified by a point (x0,y0,z0)(x_0, y_0, z_0) on the line, and a direction vector (l,m,n)(l, m, n) of the line.

The direction vector can be replaced by any non-zero scalar multiple. The point can be replaced by any other point on the line.

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Vector equation of a line

The position vector r\bm r of any point on the line is

r=(l,m,n)λ+(x0,y0,z0)\bm r = (l, m, n)\lambda + (x_0, y_0, z_0)

The vector multiplied by a parameter λ\lambda is the direction vector.

Different points on the line are associated with different parameters.

Parametric equation of a line

x=lλ+x0x = l\lambda + x_0
y=mλ+y0y = m\lambda + y_0
z=nλ+z0z = n\lambda + z_0

Symmetric (Cartesian) equation of a line

This solves for λ\lambda and equate the expressions.

x−x0l=y−y0m=z−z0n\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}

Where the component of the direction vector is 00, for example if direction vector is (2,0,−3)(2, 0, -3), then

x−x02=z0−z3,y=y0\frac{x - x_0}{2} = \frac{z_0 - z}{3}, y = y_0

Note, when there is not a denominator, it is 11. Beware of potential of a reversed numerator, because of multiplying by −1-1.

Checking if a point is on a line

Suppose we have

r^=(3,−1,−4)λ+(2,−5,6)\bm{\hat r} = (3, -1, -4)\lambda + (2, -5, 6)

We want to verify if (−7,−2,6)(-7, -2, 6) is on the line.

If the direction vector has all non-zero components, then the parameter must be the same value for xx, yy, zz.

x:λ=−7−(2)3=−3x: \lambda = \frac{-7 - (2)}{3} = -3
y:λ=−2−(−5)−1=−3y: \lambda = \frac{-2 - (-5)}{-1} = -3
z:λ=6−(6)−4=0≠−3z: \lambda = \frac{6- (6)}{-4} = 0\neq -3

So (−7,−2,6)(-7, -2, 6) is not on the line. Note that we simply substituted values into the symmetric (Cartesian) equation of a line.

Finding equation of a line.

If given two points (x0,y0,z0)(x_0, y_0, z_0) and (x1,y1,z1)(x_1, y_1, z_1), the equation of the line is

r=(x1−x0,y1−y0,z1−z0)λ+(x0,y0,z0)\bm r = (x_1 - x_0, y_1 - y_0, z_1 - z_0)\lambda + (x_0, y_0, z_0)

Namely, find a direction vector as the difference between two points, then include one of the given points (position vectors).

Applications

The equations of a 3D line can be interpreted to represent the path of some object or particle, with initial position (x0,y0,z0)(x_0, y_0, z_0), constant velocity (l,m,n)(l, m, n), time traveled λ\lambda, and final position r\bm r.

Tips

  1. All forms of a line (using vectors) are accepted. Vector equation is probably the easiest.
  2. Use different parameters for different lines.