This is a hugely important topic in the HL course. We can describe planes in 3D in a number of different ways. It is not enough to learn the different equations, but it is vital that you have a strong conceptual understanding of the different forms of the equations of the plane, in particular, the normal form. You will need to be familiar with the Vector Product and the Scalar Product before you start this topic. Questions in the exam on this topic are often long questions from section B.
Key Concepts
On this page, you should learn about
vector equations of planes in three dimensions in the three different forms
vector form
parametric form
cartesian form
Essentials
The following videos will help you understand all the concepts from this page
Planes - Vector, Normal & Cartesian Forms
In the following video we are going to look at the equations of planes, in particular the three different forms: Vector, Normal and Cartesian.
r = a + λb + μc
Vector Form
r∙n = a∙n
Normal Form
ax + by + cz = d
Cartesian Form
We will find out how to convert from one form to another by looking at the follwing example:
Convert the following into normal and Cartesian form
In the following video we are going to try to find the equation of a plane that is formed by three points. Also, we will be able to see if a 4th point lies in the same plane (Are the four points coplanar?)
Find the equation of the plane formed by the triangle A(1 , 2 , -1) , B(2 , -2 , 3) and C(0 , 2 , 1)
Which of the following is the equation of a plane?
Find all correct answers
\(\frac { x-2 }{ 3 } =\frac { y+5 }{ -2 } =z\) and \(\textbf{ r }=\left( \begin{matrix} 2 \\ -1 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \) are equations of planes in 3D
Which of the following is a point that lies on the plane \(\textbf{ r }=\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} -2 \\ 0 \\ 3 \end{matrix} \right) \) ?
The vector form of a plane is given in the form r=a+λb+μc where a is a point that lies in the plane.
Therefore the position vector \(\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right)\) is a point on the plane
Which of the following is a point that lies on the plane
x = 1 + 2λ - 2μ
y = -1 - 1λb + 4μ
z = 4 + 3λb
This is the parametric form of the equation of a plane. Try writing it in vector form.
\(\left( \begin{matrix} a \\ b \\ c \end{matrix} \right) \)is a vector perpendicular to the plane x + 2y - 3z = 8. Find a, b and c.
a =
b =
c =
The plane here is given in Cartesian form. The coefficients of x, y and z give us the normal to the vector. Any multiple of the vector \(\left( \begin{matrix} 1 \\ 2 \\ -3 \end{matrix} \right) \)is perpendicular to the plane.
Which of the following vectors is a normal to the plane
Show that the Cartesian equation of the plane is x - 5y + 3z + 9 = 0
Hint
Find the normal to the plane by finding the vector product of \(\left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right)\) and \(\left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)
This is a hugely important topic in the HL course. We can describe planes in 3D in a number of different ways. It is not enough to learn the different equations, but it is vital that you have a strong conceptual understanding of the different forms of the equations of the plane, in particular, the normal form. You will need to be familiar with the Vector Product and the Scalar Product before you start this topic. Questions in the exam on this topic are often long questions from section B.
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