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1

Find the equation of a plane from a point and a normal vector

e.g. “ Find the equation of the plane that passes through the point \dots and has normal vector \dots ”

1
$\mathbf n = \langle a, b, c\rangle$ $A$
2
$(x_0, y_0, z_0)$ $A$
3
$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
4
$ax + by + cz = d$

Remember: Normal from the coefficients, point for the constant.

See it in a problem4 worked examples

(2, -1, 4)

1
$\mathbf n = \langle a, b, c\rangle$ $\mathbf n = \langle 3, 1, -2\rangle$
2
$(x_0, y_0, z_0)$ $(2, -1, 4)$
3
$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$ $3(x - 2) + (y + 1) - 2(z - 4) = 0$
4
$ax + by + cz = d$ $3x + y - 2z = -3$

3x + y - 2z = -3

Common mistakes3

Signs in $a(x - x_0)$ : the point $(2, -1, 4)$ gives $3(x - 2) + (y + 1) - 2(z - 4)$ .
Using a direction in the plane as its normal. The normal is perpendicular to the plane.
Calling the normal "the plane". The answer is an equation in $x$ , $y$ , $z$ .

2

Find the equation of a plane through three points

e.g. “ Find the plane containing the given three points ”

1
$Find two non-parallel vectors that lie in the plane$ $P, Q, R$
2
$\mathbf n = \mathbf u\times\mathbf v$ $\mathbf n = \mathbf 0$
3
$ax + by + cz = \mathbf n\cdot P$
4
$Check the other points$

Remember: Two vectors in the plane, cross, plug in.

See it in a problem3 worked examples

P = (1, 2, 0)

1
$Find two non-parallel vectors that lie in the plane$ $\overrightarrow{PQ} = \langle 2, -1, 1\rangle$
2
$\mathbf n = \mathbf u\times\mathbf v$ $\overrightarrow{PQ}\times\overrightarrow{PR} = \langle -2 + 1,\ -1 - 4,\ -2 - 1\rangle = \langle -1, -5, -3\rangle$
3
$ax + by + cz = \mathbf n\cdot P$ $x + 5y + 3z = 1 + 10 + 0 = 11$
4
$Check the other points$ $Q$

x + 5y + 3z = 11

Common mistakes3

Crossing the points themselves, like $\langle 1, 2, 0\rangle\times\langle 3, 1, 1\rangle$ , instead of vectors between points.
Plane through a point and a line: the line's point alone is not a second vector. Use $A - P$ , from the line to the given point.
Skipping the check: one slipped sign in the cross product gives a plane that misses a point.

3

Find the distance from a point to a plane

e.g. “ Find the distance from the given point to the given plane ”

1
$ax + by + cz = d$ $Only given points? Find the plane first (three-points recipe).$
2
$ax_0 + by_0 + cz_0 - d$ $a, b, c$
3
$|\mathbf n|$

Remember: Plug in, divide by the normal's length.

See it in a problem3 worked examples

(3, -1, 2)

1
$ax + by + cz = d$ $\mathbf n = \langle 2, -1, 2\rangle$
2
$ax_0 + by_0 + cz_0 - d$ $6 + 1 + 4 - 1 = 10$
3
$|\mathbf n|$ $|\mathbf n| = 3$

\dfrac{10}{3}

Common mistakes3

Parallel planes with scaled normals: rewrite $-2x + 4y - 4z = 10$ as $x - 2y + 2z = -5$ before subtracting.
Dividing by $a^2 + b^2 + c^2$ instead of its square root.
"All planes at distance $2$ ": there are two, one on each side.

4

Find where a line intersects a plane

e.g. “ Find the point where the line \dots meets the plane \dots ”

1
$x(t)$ $D$
2
$Substitute into the plane's equation$
3
$t$ $t$
4
$t$

Remember: Substitute, solve for t, plug back.

See it in a problem3 worked examples

\langle 1, 2, -1\rangle + t\langle 2, -1, 3\rangle

1
$x(t)$ $x = 1 + 2t$
2
$Substitute into the plane's equation$ $(1 + 2t) + (2 - t) + (-1 + 3t) = 8$
3
$t$ $2 + 4t = 8$
4
$t$ $\left(1 + 3,\ 2 - \tfrac{3}{2},\ -1 + \tfrac{9}{2}\right) = \left(4, \tfrac{1}{2}, \tfrac{7}{2}\right)$

\left(4, \tfrac{1}{2}, \tfrac{7}{2}\right)

Common mistakes3

Stopping at $t$ : the question asks for a point.
A shadow from a light at $O$ : the line runs from $O$ through the point, so its direction is the point itself.
Thinking " $2 = 5$ " must be a slip. If your algebra is right, it means the line never meets the plane.

Equations of Planes

Find the equation of a plane from a point and a normal vector

Find the equation of a plane through three points

Find the distance from a point to a plane

Find where a line intersects a plane

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Navigation

Find the equation of a plane from a point and a normal vector

Find the equation of a plane through three points

Find the distance from a point to a plane

Find where a line intersects a plane