Vector and Cartesian equations of planesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

Vector equation of a plane?
How many parameters does a plane need?
Directions of the plane through A, B, C?
What is a normal vector to a plane?
How do you find a normal from directions \mathbf b, \mathbf c?
Scalar product form of a plane?
Cartesian equation of a plane?
What do a, b, c represent in ax+by+cz=d?
Cartesian form of \mathbf r\cdot(2,-1,3)=8?
How do you test if a point is on a plane?
How can you recognise parallel planes?
Direction of a line perpendicular to a plane?

Exam questions on Vector and Cartesian equations of planes

The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$ .
Determine whether the point $(3,4,2)$ lies on $\Pi$ .2 marks
The points $A(1,2,0)$ , $B(3,2,1)$ and $C(2,5,2)$ lie in a plane $\Pi$ .
Hence find a Cartesian equation of $\Pi$ .2 marks
The plane $\Pi$ has Cartesian equation $3x-2y+z=12$ .
Find a vector equation of $\Pi$ in the form $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).