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Here are some exam-style questions on this statement:
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if a plane has normal vector \( \mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) and passes through \( (X,Y,Z) \) then the Cartesian equation of the plane is:
$$ ax + by + cz = aX + bY + cZ$$This is derived from the equation in the formula booklet:
$$ r \cdot n=a \cdot n $$E.g. If a plane passes through (1,2,3) and has a normal \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} then the equation of the plane is found by simplifying this:
$$ \begin{pmatrix} x \\ y \\ x \end{pmatrix} \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} $$If the Cartesian equation of a plane is \(ax+by+cz=d\) then the vector normal to the plane is:
$$ \mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.
