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International Baccalaureate Mathematics

Geometry and Trigonometry

Syllabus Content

Vector equations of a plane: r=a+λb+μc, where b and c are non-parallel vectors within the plane.
r·n=a·n, where n is a normal to the plane and a is the position vector of a point on the plane.
Cartesian equation of a plane ax+by+cz=d.

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Furthermore

Formula Booklet:

Vector equation of a plane

Equation of a plane (using the normal vector)

Cartesian equation of a plane

\(r=a + \lambda b+ \mu c\)

\( r \cdot n=a \cdot n \)

\( ax + by + cz = d \)

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:

$$ \mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$

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