AHL 3.17—Vector equations of a plane Questionbank

Practice AHL 3.17—Vector equations of a plane with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the plane $\Pi_1$ given by the equation $x + 2y - z = 4$ and the plane $\Pi_2$ given by the equation $3x - y + 4z = 10$ .

Find a vector equation of the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify that the line of intersection found is perpendicular to the normal vectors of both planes.

[3]

Question 2

HLPaper 2

The points $A(5, -2, 5)$ , $B(5, 4, -1)$ , $C(-1, -2, -1)$ and $D(7, -4, -3)$ are the vertices of a right pyramid. The line $L$ passes through the point $D$ and is perpendicular to plane $\Phi$ .

Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[2]

Show that the Cartesian equation of the plane $\Phi$ that contains the triangle $ABC$ is $-x+y+z=-2$ .

[3]

Find a vector equation of the line $L$ .

[1]

Hence determine the minimum distance, $d_{\text{min}}$ , from $D$ to $\Phi$ .

[4]

Use a vector method to show that $B\hat{A}C = 60^\circ$ .

[3]

Find the volume of right pyramid $ABCD$ .

[4]

Question 3

HLPaper 2

Consider two planes $\Pi_1$ and $\Pi_2$ with equations:

\Pi_1: 2x + 3y - z = 4

\Pi_2: -x + 2y + 2z = 6

Find the angle between the two planes.

[3]

Find a point that lies on both planes.

[3]

Find a direction vector of the line of intersection of these planes.

[3]

Question 4

HLPaper 1

Consider the line $L_1$ given by the vector equation $\mathbf{r} = \begin{pmatrix} 8 \\ 4 \\ 9 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ 9 \end{pmatrix}$ and the plane $\pi$ given by the equation $2x - y + 3z = 12$ .

Find the point of intersection between the line $L_1$ and the plane $\pi$ .

[3]

Determine if the line $L_1$ is parallel, perpendicular, or neither to the plane $\pi$ .

[4]

Question 5

HLPaper 1

Consider the line $L_1$ given by the equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\pi$ given by the equation $2x - y + z = 5$ .

Find the point of intersection of the line $L_1$ and the plane $\pi$ .

[4]

Determine if the line $L_1$ is parallel, perpendicular, or neither to the normal of the plane $\pi$ .

[4]

Question 6

HLPaper 1

Find the exact value of cosine of the angle $\theta$ between the planes $\pi_{1}$ and $\pi_{2}$ , where $\pi_{1}$ has equation $-2 x+y-z=2$ and $\pi_{2}$ has equation $x+2 y-z=6$

[8]

Question 7

HLPaper 1

The points $A, B, C$ have position vectors $\mathbf{i}+\mathbf{j}+2\mathbf{k}$ , $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ , and $3\mathbf{i}+\mathbf{k}$ respectively and lie in the plane $\pi$ .

Find the area of the triangle $ABC$ .

[6]

Hence find the shortest distance from $C$ to the line $AB$ .

[5]

Find the cartesian equation of the plane $\pi$ .

[5]

Question 8

HLPaper 1

Let the line $L$ and plane $\Pi$ be given by
$L:\ \mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+t\begin{pmatrix}1\\2\\2\end{pmatrix},\\ \Pi:\ 2x-y+2z=5.$

Show that $L$ is not contained in $\Pi$ .

[3]

Find the point of intersection of $L$ and $\Pi$ .

[3]

Find the distance from the point $P(5,0,2)$ to the line $L$ .

[4]

Find the shortest distance between the line $L$ and the plane $\Pi_1:\ 2x-y+2z=1.$

[3]

Hence determine the acute angle between line $L$ and plane $\Pi$ .

[3]

Question 9

HLPaper 2

Consider the planes $\pi_{1}: 2x + 3y + z = 7$ and $\pi_{2}: 4x + 7y + 4z = 19$ . Find the following:

The acute angle between the planes $\pi_{1}$ and $\pi_{2}$
The acute angle between the $y$ -axis and $\pi_{2}$

The acute angle between the planes $\pi_{1}$ and $\pi_{2}$

[9]

The acute angle between the $y$ -axis and $\pi_{2}$

[5]

Question 10

HLPaper 1

Consider a line $L$ and a plane $P$ in a three-dimensional space.

The line $L$ is given by the parametric equations $x = 2 + t$ , $y = 3 - 2t$ , $z = 1 + 4t$ . The plane $P$ is given by the equation $2x - y + z = 5$ . Find the point of intersection of the line $L$ and the plane $P$ .

[3]

Determine whether the line $L$ is parallel, perpendicular, or neither to the normal vector of the plane $P$ .

[3]