Equation of a Plane

MATPlanes

A plane is fixed by a point on it and a normal direction. Standard forms include the normal-vector form, the general Cartesian form $ax+by+cz+d=0$ , the three-point form via a determinant, and the family of planes through the intersection of two planes.

ISC examines writing the plane through a point with a given normal, through three points, and through a line of intersection passing through a given point.

Plane through a point with normal vector (vector form)

(\vec{r} - \vec{a}) \cdot \vec{n} = 0

\vec{a}

is the position vector of a known point,

\vec{n}

is the normal vector; equivalently

\vec{r}\cdot\vec{n} = \vec{a}\cdot\vec{n}

Cartesian plane through a point with normal

(a,b,c)

a(x - x_1) + b(y - y_1) + c(z - z_1) = 0

(x_1,y_1,z_1)

lies on the plane and

(a,b,c)

are the direction ratios of the normal; the general form is

ax+by+cz+d=0

Plane through three points (determinant form)

\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0

A(x_1,y_1,z_1), B(x_2,y_2,z_2), C(x_3,y_3,z_3)

are three non-collinear points on the plane.

Intercept form of a plane

\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1

a, b, c

are the intercepts the plane makes on the

x

y

- and

z

-axes respectively.

Plane through line of intersection of two planes

(a_1 x + b_1 y + c_1 z + d_1) + \lambda(a_2 x + b_2 y + c_2 z + d_2) = 0

The two given planes are

P_1 = 0

and

P_2 = 0

;

\lambda

is determined by an extra known condition (a point on, or a property of, the required plane).

The coefficients $(a,b,c)$ in $ax+by+cz+d=0$ are exactly the direction ratios of the plane's normal vector $\vec{n}$ .
For three points, the normal is $\vec{n} = \overrightarrow{AB} \times \overrightarrow{AC}$ ; then use the point-normal form — equivalent to the determinant.
A plane perpendicular to a given vector has that vector as its normal; a plane parallel to a given plane shares the same normal coefficients (only $d$ changes).
To make a plane contain a given line, the line's direction must be perpendicular to the plane's normal ( $\vec{b}\cdot\vec{n}=0$ ) AND a point of the line must satisfy the plane.
In the family form, $\lambda$ is determined by one extra condition (a point on the plane, perpendicularity, or a given normal direction).
Convert vector form $\vec{r}\cdot\vec{n}=d$ to Cartesian by writing $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and taking the dot product.
The normal (perpendicular) form $\vec{r}\cdot\hat{n} = p$ uses the unit normal $\hat{n}$ , where $p \ge 0$ is the perpendicular distance of the plane from the origin.

Where the marks go

Choosing a normal that is not perpendicular to the plane (e.g. using a line's direction as the normal when the plane should contain that line).
Sign or arithmetic slips while expanding the $3\times 3$ determinant in the three-point form.
In the family-of-planes method, forgetting to substitute the extra condition to solve for $\lambda$ , or omitting the $P_2$ term entirely.
Confusing 'parallel to a plane' (same normal) with 'perpendicular to a plane' (normal lies in the required plane).

How the board asks it

Numericalpoint-normal form of a plane
Find the vector and Cartesian equations of the plane passing through the point $(1,-2,3)$ and perpendicular to the vector $2\hat{i}+3\hat{j}-4\hat{k}$ .
Numericalplane through three points
Find the equation of the plane passing through the three points $A(2,1,-1)$ , $B(-1,3,4)$ and $C(0,-2,1)$ .
Numericalplane through line of intersection of two planes
Find the equation of the plane through the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ which passes through the point $(1,0,-2)$ .
Numericalplane parallel to a given plane (same normal, only $d$ changes)
Find the equation of the plane passing through the point $(2,-1,4)$ and parallel to the plane $3x-2y+6z+7=0$ .
Conversionvector form to Cartesian via dot product
Reduce the equation of the plane $\vec{r}\cdot(2\hat{i}-3\hat{j}+4\hat{k})=6$ to Cartesian form, and hence write the direction ratios of its normal.
Numericalplane containing a line (normal perpendicular to the line)
Find the equation of the plane containing the line $\dfrac{x-1}{2}=\dfrac{y+1}{-1}=\dfrac{z}{3}$ and passing through the point $(0,1,2)$ .

Practise this topic

PreviousAngle Between Two Lines & Shortest Distance Between Skew Lines NextAngle Between Planes / Line and Plane, and Distance of a Point from a Plane

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.