A plane: parametric form and equation

Vectors, grade 12 (5 units) Lesson 17 of 23Learning

A plane: parametric form and equation

Parametric form

A plane through the point $\displaystyle P$ , with two independent vectors $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ parallel to it: every vector in the plane is a linear combination of them, so every point of the plane is

\displaystyle \underline{x}

\displaystyle =

\displaystyle \underline{p}+t\cdot\underline{u}+s\cdot\underline{v}

with two parameters, $\displaystyle t$ and $\displaystyle s$ . The plane through three points $\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ not on one line: $\displaystyle {\underline{x}}={A+t(B-A)+s(C-A)}$ .

The normal vector and the equation

A vector $\displaystyle {\underline{n}}={(a,b,c)}$ perpendicular to a plane is called its normal vector. The point $\displaystyle X(x,y,z)$ is in the plane through $\displaystyle P(x_1,y_1,z_1)$ if and only if $\displaystyle \overrightarrow{PX}$ lies in the plane, that is $\displaystyle {\underline{n}\cdot\overrightarrow{PX}}={0}$ :

\displaystyle a(x-x_1)+b(y-y_1)+c(z-z_1)

\displaystyle =

\displaystyle 0

After opening the brackets: $\displaystyle {ax+by+cz+d}={0}$ . The coefficients of $\displaystyle x$ , $\displaystyle y$ , $\displaystyle z$ are the normal – for example, the normal of $\displaystyle {2x-y+3z}={6}$ is $\displaystyle (2,-1,3)$ .

Finding the normal

From parametric form to an equation: the normal is perpendicular to $\displaystyle \underline{u}$ and to $\displaystyle \underline{v}$ .

$\displaystyle {\underline{n}\cdot\underline{u}}={0}$ and $\displaystyle {\underline{n}\cdot\underline{v}}={0}$ : two equations in three unknowns $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ .
Every multiple of a normal is a normal, so one unknown may be chosen (for example $\displaystyle {c}={1}$ ) and the other two solved.
Put $\displaystyle \underline{n}$ and a point of the plane in $\displaystyle {a(x-x_1)+b(y-y_1)+c(z-z_1)}={0}$ .

Example: a plane through three points

Find the equation of the plane through $\displaystyle A(1,0,0)$ , $\displaystyle B(0,2,0)$ , $\displaystyle C(0,0,3)$ .

$\displaystyle {\overrightarrow{AB}}={(-1,2,0)}$ , $\displaystyle {\overrightarrow{AC}}={(-1,0,3)}$ .
$\displaystyle {\underline{n}\cdot\overrightarrow{AB}}={-a+2b}={0}$ , $\displaystyle {\underline{n}\cdot\overrightarrow{AC}}={-a+3c}={0}$ . Choose $\displaystyle {a}={6}$ : $\displaystyle {b}={3}$ , $\displaystyle {c}={2}$ .
$\displaystyle {6(x-1)+3y+2z}={0}$ , that is $\displaystyle {6x+3y+2z}={6}$ .
Check: $\displaystyle B$ gives $\displaystyle 6$ , and $\displaystyle C$ gives $\displaystyle 6$ .

Where it cuts the axes: on the $\displaystyle x$ -axis – $\displaystyle {y}={z}={0}$ ; on the $\displaystyle y$ -axis – $\displaystyle {x}={z}={0}$ ; on the $\displaystyle z$ -axis – $\displaystyle {x}={y}={0}$ .

Special planes

$\displaystyle {z}={0}$ – the $\displaystyle xy$ plane; $\displaystyle {z}={3}$ – a plane parallel to it, at height 3. The normal is $\displaystyle (0,0,1)$ .
$\displaystyle {x+y}={4}$ – there is no $\displaystyle z$ in it: the normal $\displaystyle (1,1,0)$ is perpendicular to the $\displaystyle z$ -axis, so the plane is parallel to the $\displaystyle z$ -axis.
$\displaystyle {ax+by+cz}={0}$ (no free number) – the plane passes through the origin.

Exercise 1

Write the equation of the plane through $\displaystyle A(1,1,1)$ whose normal vector is $\displaystyle (2,-1,3)$ , in the form $\displaystyle {ax+by+cz}={d}$ .

Exercise 2

Find the equation of the plane through $\displaystyle A(1,0,2)$ , $\displaystyle B(2,1,0)$ , $\displaystyle C(0,3,1)$ .

\displaystyle \overrightarrow{AB}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle \overrightarrow{AC}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle {\underline{n}}={(a,b,1)}$ . Write $\displaystyle {\underline{n}\cdot\overrightarrow{AB}}={0}$ and $\displaystyle {\underline{n}\cdot\overrightarrow{AC}}={0}$ , and solve.

The plane in parametric form – write its equation.

Exercise 3

The plane $\displaystyle {2x+3y+6z}={12}$ .

At which points does it cut the axes?
Write its normal vector, and two vectors parallel to it.

Exercise 4

Is the point $\displaystyle (1,2,3)$ in the plane $\displaystyle {2x-y+3z}={9}$ ? Put it in.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: vectors, grade 12 (5 units)

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Solids in space

The volume of a prism and a cylinder – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral area of a right cone – $\displaystyle R$ the base’s radius, $\displaystyle \ell$ the slant height
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\ell$
The volume of a sphere – $\displaystyle R$ its radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^3$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^2$
The diagonal of a box of dimensions $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2+c^2}$

Trigonometry in the triangle

The law of sines – $\displaystyle R$ the radius of the circumscribed circle
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$
The area of a triangle – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

Vectors in the geometric approach

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
A vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{OB}-\overrightarrow{OA}$
Collinear (parallel) vectors
$\displaystyle \underline{v}=t\cdot\underline{u}$
The point $\displaystyle M$ divides the segment $\displaystyle AB$ in the ratio $\displaystyle m:n$
$\displaystyle \overrightarrow{OM}=\frac{n}{m+n}\overrightarrow{OA}+\frac{m}{m+n}\overrightarrow{OB}$ $\displaystyle AM:MB=m:n$
$\displaystyle M$ the midpoint of the segment $\displaystyle AB$
$\displaystyle \overrightarrow{OM}=\frac{1}{2}\left(\overrightarrow{OA}+\overrightarrow{OB}\right)$

The scalar product

The length of a vector $\displaystyle {\underline{u}}={(u_1,u_2,u_3)}$
$\displaystyle |\underline{u}|=\sqrt{\underline{u}\cdot\underline{u}}=\sqrt{u_1^2+u_2^2+u_3^2}$
The scalar product in coordinates
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle u_1v_1+u_2v_2+u_3v_3$
The scalar product – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle |\underline{u}|\cdot|\underline{v}|\cdot\cos\alpha$
Perpendicular vectors
$\displaystyle \underline{u}\perp\underline{v}\iff\underline{u}\cdot\underline{v}=0$
A short scalar product
$\displaystyle (\underline{u}+\underline{v})^2$ $\displaystyle =$ $\displaystyle \underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2$

Points and vectors in axes

A vector between two points
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
The distance between two points
$\displaystyle AB$ $\displaystyle =$ $\displaystyle \sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$
The midpoint of the segment $\displaystyle AB$
$\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)$

The line and the plane

A line – $\displaystyle \underline{v}$ a point on the line, $\displaystyle \underline{u}$ the direction vector
$\displaystyle \underline{x}=\underline{v}+t\cdot\underline{u}$
A plane in parametric form – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ not parallel
$\displaystyle \underline{x}=\underline{p}+t\cdot\underline{u}+s\cdot\underline{v}$
A plane’s equation – $\displaystyle {\underline{n}}={(a,b,c)}$ the normal vector
$\displaystyle ax+by+cz+d=0$
The plane through $\displaystyle P(x_1,y_1,z_1)$ whose normal vector is $\displaystyle (a,b,c)$
$\displaystyle a(x-x_1)+b(y-y_1)+c(z-z_1)=0$

Distances

The distance of a point $\displaystyle P(x_1,y_1,z_1)$ from the plane $\displaystyle {ax+by+cz+d}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|a\cdot x_1+b\cdot y_1+c\cdot z_1+d|}{\sqrt{a^2+b^2+c^2}}$
The distance between the parallel planes $\displaystyle {ax+by+cz+d_1}={0}$ and $\displaystyle {ax+by+cz+d_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$

Angles

The angle between two lines – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ the direction vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{u}\cdot\underline{v}|}{|\underline{u}|\cdot|\underline{v}|}$
The angle between the line $\displaystyle {\underline{x}}={\underline{v}+t\cdot\underline{u}}$ and the plane whose normal is $\displaystyle \underline{n}$
$\displaystyle \sin\beta$ $\displaystyle =$ $\displaystyle \frac{|\underline{n}\cdot\underline{u}|}{|\underline{n}|\cdot|\underline{u}|}$
The angle between two planes – $\displaystyle \underline{n_1}$ , $\displaystyle \underline{n_2}$ the normals
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{n_1}\cdot\underline{n_2}|}{|\underline{n_1}|\cdot|\underline{n_2}|}$