Equations of planes (Edexcel A Level Further Maths: Core Pure): Revision Note

Equation of a plane in vector form

How do I find the vector equation of a plane?

• A plane is a flat surface which is two-dimensional

  • Imagine a flat piece of paper that continues on forever in both directions

• A plane in often denoted using the capital Greek letter Π

• The vector form of the equation of a plane can be found using two direction vectors on the plane

  • The direction vectors must be

    • parallel to the plane

    • not parallel to each other

    • therefore they will intersect at some point on the plane

• The formula for finding the vector equation of a plane is

  • $\mathit{r}=\mathit{a}+s\mathit{b}+t\mathit{c}$

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

    • a is the position vector of a known point on the plane

    • b and c are two non-parallel direction (displacement) vectors parallel to the plane

    • s and t are scalars

• The formula can also be written as

  • $\mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a})+\mu (\mathbf{c}-\mathbf{a})=(1-\lambda -\mu )\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c}$

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

    • a, b, c are the position vectors of known points on the plane

    • λ and μ are scalars

  • These formulae are given in the formula booklet but you must make sure you know what each part means

• As a could be the position vector of any point on the plane and b and c could be any non-parallel direction vectors on the plane there are infinite vector equations for a single plane

How do I determine whether a point lies on a plane?

• Given the equation of a plane $\mathit{r}\mathit{ }=\mathit{ }\left(\genfrac{}{}{0pt}{}{{\mathit{a}}_1}{\begin{array}{c}{\mathit{a}}_2\\ {\mathit{a}}_3\end{array}}\right)+\lambda \left(\genfrac{}{}{0pt}{}{{\mathit{b}}_1}{\begin{array}{c}{\mathit{b}}_2\\ {\mathit{b}}_3\end{array}}\right)+ \mu \left(\genfrac{}{}{0pt}{}{{\mathit{c}}_1}{\begin{array}{c}{\mathit{c}}_2\\ {\mathit{c}}_3\end{array}}\right)$ then the point r with position vector $\mathit{ }\left(\genfrac{}{}{0pt}{}{x}{\begin{array}{c}y\\ z\end{array}}\right)$ is on the plane if there exists a value of λ and μ such that

  • $\left(\genfrac{}{}{0pt}{}{x}{\begin{array}{c}y\\ z\end{array}}\right)\mathrm{ }=\mathit{ }\left(\genfrac{}{}{0pt}{}{{\mathit{a}}_1}{\begin{array}{c}{\mathit{a}}_2\\ {\mathit{a}}_3\end{array}}\right)+\lambda \left(\genfrac{}{}{0pt}{}{{\mathit{b}}_1}{\begin{array}{c}{\mathit{b}}_2\\ {\mathit{b}}_3\end{array}}\right)+ \mu \left(\genfrac{}{}{0pt}{}{{\mathit{c}}_1}{\begin{array}{c}{\mathit{c}}_2\\ {\mathit{c}}_3\end{array}}\right)$

  • This means that there exists a single value of λ and μ that satisfy the three parametric equations:

    • $x= a_1+\lambda b_1+\mathrm{ }\mu c_1\mathrm{ }$

    • $y= a_2+\lambda b_2+\mathrm{ }\mu c_2\mathrm{ }$

    • $z= a_3+\lambda b_3 +\mathrm{ }\mu c_2$

• Solve two of the equations first to find the values of λ and μ that satisfy the first two equation and then check that this value also satisfies the third equation

• If the values of λ and μ do not satisfy all three equations, then the point r does not lie on the plane

Equation of a plane in Cartesian form

How do I find the vector equation of a plane in cartesian form?

• The cartesian equation of a plane is given in the form

  • $ax+by+cz=d$

  • This is given in the formula booklet

• A normal vector to the plane can be used along with a known point on the plane to find the cartesian equation of the plane

  • The normal vector will be a vector that is perpendicular to the plane

• The scalar product of the normal vector and any direction vector on the plane will be zero

  • The two vectors will be perpendicular to each other

  • The direction vector from a fixed-point A to any point on the plane, R can be written as r – a

  • Then n ∙ (r – a) = 0 and it follows that (n ∙ r) – (n ∙ a) = 0

• This gives the equation of a plane using the normal vector:

  • n ∙ r = a ∙ n

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

    • a is the position vector of a known point on the plane

    • n is a vector that is normal to the plane

  • This is given in the formula booklet

• If the vector r is given in the form $\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$ and a and n are both known vectors given in the form $\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right)$ and $\left(\begin{array}{c}{n}_1\\ n_2\\ n_3\end{array}\right)$ then the Cartesian equation of the plane can be found using:

  • $\mathit{n}·\mathit{r}=n_{1}x+n_{2}y+n_{3}z$

  • $\mathit{a}·\mathit{n}=a_1{n}_1+a_2{n}_2+a_3{n}_3$

  • Therefore $n_{1}x+n_{2}y+n_{3}z=a_1{n}_1+a_2{n}_2+a_3{n}_3$

  • This simplifies to the form $ax+by+cz=d$

    • A version of this is given in the formula booklet

How do I find the equation of a plane in Cartesian form given the vector form?

• Given the equation of the plane $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c}$

  • Form three equations

    • $x=a_1+\lambda b_1+\mu c_1$

    • $y=a_2+\lambda b_2+\mu c_2$

    • $z=a_3+\lambda b_3+\mu c_3$

• Choose a pair of equations and use them to form an equation without μ

• Choose another pair and form another equation without μ

• Use your two expressions to form an equation without μ and λ

• Rewrite the equation in the form $ax+by+cz+d=0$