IB Mathematics Analysis and Approaches Topic AHL 3.14—vector Equation of Line Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 3.14—vector Equation of Line Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 3.14—vector Equation of Line with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 3.14—vector Equation of Line and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Two lines in three-dimensional space are given by $L_1:\ \mathbf r = \begin{pmatrix} 0\\ 1\\ 2 \end{pmatrix} +\lambda \begin{pmatrix} 1\\ -1\\ 2 \end{pmatrix}, \qquad L_2:\ \mathbf r = \begin{pmatrix} 2\\ 3\\ 0 \end{pmatrix} +\mu \begin{pmatrix} 2\\ -2\\ 4 \end{pmatrix}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[4]

Find the acute angle between the lines.

[2]

Let $S = (1, 0, 0)$ . Find the shortest distance from $S$ to line $L_{1}$ , and the coordinates of the foot $F$ of the perpendicular.

[5]

Let $P = (2, 3, 0)$ , a point on $L_{2}$ . Find all points $Q$ on $L_{2}$ such that $P Q$ is perpendicular to the direction of $L_{1}$ .

[4]

Question 2

HLPaper 1

The line $L$ is given by the parametric equations $x = 1 - λ, y = 2 - 3 λ, z = 2$ .

Find the coordinates of the point on $L$ that is nearest to the origin.

[4]

Question 3

HLPaper 2

Two lines are given by $L_1:\ \mathbf r = \begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix} +\lambda \begin{pmatrix} 2\\ -1\\ 2 \end{pmatrix}, \qquad L_2:\ \mathbf r = \begin{pmatrix} 5\\ -1\\ 3 \end{pmatrix} +\mu \begin{pmatrix} -2\\ 1\\ 1 \end{pmatrix}.$

Write both lines in parametric and Cartesian form.

[3]

Determine the relationship between the lines and, if they intersect, find the point of intersection.

[4]

Find the acute angle between $L_{1}$ and $L_{2}$ .

[4]

Let $S = (3, 1, 0)$ . Find the shortest distance from $S$ to $L_{1}$ , and the coordinates of the foot $F$ .

[5]

Question 4

HLPaper 1

Two lines in three-dimensional space are given by $L_1: \mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2: \mathbf{r}=\begin{pmatrix}5\\0\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix},\qquad \lambda,\mu\in\mathbb{R}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[3]

Find the acute angle $θ$ between $L_{1}$ and $L_{2}$ , giving an exact value for $cos θ$ and $θ$ (in degrees).

[3]

Let $S = (2, 0, 5)$ . Find the shortest distance from $S$ to the line $L_{1}$ , and the coordinates of the foot $F$ of the perpendicular from $S$ to $L_{1}$ .

[4]

Find all points $Q$ on $L_{2}$ such that the vector $P Q$ makes an angle of $4 5^{\circ}$ with the direction of $L_{1}$ , where $P$ is the intersection point from Part 2. Give exact parameter values $μ$ and coordinates of $Q$ if they exist.

[5]

Question 5

HLPaper 2

In this question, distances are in km and times in hours.

Airplane $1$ passes through point $P (2, 5, 8)$ at time $t = 0$ with constant velocity $v_{1} = (4, - 2, 1)$ . Airplane $2$ passes through point $Q (0, 0, 10)$ at time $t = 0$ with constant velocity $v_{2} = (3, 1, - 2)$ .

Find the speed of each airplane.

[4]

Write down the position vector of:

(i) airplane $1$ at time $t$ ;

(ii) airplane $2$ at time $t$ .

[3]

Show that the two airplanes do not collide.

[4]

Find the time $t$ at which the two airplanes are closest.

[4]

Find the minimum distance between the airplanes.

[3]

IB Mathematics Analysis and Approaches AHL 3.14—vector Equation of Line Questionbank

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IB Mathematics Analysis and Approaches AHL 3.14—vector Equation of Line Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.14—vector Equation of Line Questionbank

Unlock all 18 questions with a free account