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AHL 3.17—Vector equations of a plane - IB Questionbank | RevisionDojo

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 the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify if the line of intersection found is perpendicular to the normal of either plane.

[3]

Question 2

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 3

HLPaper 1

Three points $A(1,2,3)$ , $B(3,-1,4)$ , and $C(2,0,6)$ define a plane $\Pi$ .

Find the vector equation of the plane $\Pi$ .

[3]

Find the Cartesian equation of $\Pi$ .

[2]

Verify that point $D(3,1,1)$ does not lie on $\Pi$ , and find the perpendicular distance from $D$ to $\Pi$ .

[3]

Find the acute angle between the plane $\Pi$ and the line
$L:\ \mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+t\begin{pmatrix}1\\-3\\-2\end{pmatrix}.$

[4]

Find a unit vector lying in the plane $\Pi$ and perpendicular to the line of intersection of $\Pi$ with the $xz$ -plane.

[3]

Question 4

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 ${\vec{AB}}$ and ${\vec{AC}}$ .

[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 5

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.

[4]

Question 6

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 7

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 8

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 9

HLPaper 1

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

Find the area of the triangle $A B C$

[6]

Hence find the shortest distance from $C$ to the line $A B$ .

[5]

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

[5]

Question 10

HLPaper 2

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

AHL 3.17—Vector equations of a plane Questionbank