Cartesian equation of a plane?

ax+by+cz=d, with normal (a,b,c).

Normal form of a plane?

\mathbf r\cdot\mathbf n=k, where k=\mathbf a\cdot\mathbf n.

Scalar product with an angle?

\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta.

How do you find a normal from \mathbf b and \mathbf c?

Solve \mathbf n\cdot\mathbf b=0 and \mathbf n\cdot\mathbf c=0.

How do you find d in ax+by+cz=d?

d=\mathbf a\cdot\mathbf n for a point \mathbf a on the plane.

Angle between two lines?

Acute angle between their direction vectors.

Angle between two planes?

Acute angle between their normals.

Angle between a line and a plane?

90^\circ minus the angle between the line and the normal, or \sin\varphi=\frac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|}.

How do you find where a line meets a plane?

Substitute the general point of the line into the plane equation and solve for the parameter.

Planes and the scalar productEdexcel A-Level Further Maths: Flashcards

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Vector equation of a plane?

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Vector equation of a plane?: $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c$ , with $\mathbf b$ , $\mathbf c$ not parallel.
Cartesian equation of a plane?: $ax+by+cz=d$ , with normal $(a,b,c)$ .
Normal form of a plane?: $\mathbf r\cdot\mathbf n=k$ , where $k=\mathbf a\cdot\mathbf n$ .
Scalar product in components?: $a_1b_1+a_2b_2+a_3b_3$ .
Scalar product with an angle?: $\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta$ .
Is the scalar product a number or a vector?: A number.
Test for perpendicular vectors?: $\mathbf a\cdot\mathbf b=0$ .
How do you find a normal from $\mathbf b$ and $\mathbf c$ ?: Solve $\mathbf n\cdot\mathbf b=0$ and $\mathbf n\cdot\mathbf c=0$ .
How do you find $d$ in $ax+by+cz=d$ ?: $d=\mathbf a\cdot\mathbf n$ for a point $\mathbf a$ on the plane.
Angle between two lines?: Acute angle between their direction vectors.
Angle between two planes?: Acute angle between their normals.
Angle between a line and a plane?: $90^\circ$ minus the angle between the line and the normal, or $\sin\varphi=\frac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|}$ .
How do you find where a line meets a plane?: Substitute the general point of the line into the plane equation and solve for the parameter.

Exam questions on Planes and the scalar product

Let $\mathbf p=3\mathbf i+2\mathbf j-\mathbf k$ and $\mathbf q=\mathbf i-4\mathbf j+2\mathbf k$ .
Find the angle between $\mathbf p$ and $\mathbf q$ , giving your answer in degrees to 1 decimal place.2 marks
The plane $\Pi_1$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$ .
The point $(k,2,4)$ lies on $\Pi_1$ . Find the value of $k$ .2 marks
The line $l$ has equation $\mathbf r=\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\Pi$ has equation $x+2y-2z=6$ .
Find the coordinates of the point where $l$ meets $\Pi$ .3 marks

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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).