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AHL 3.11—Vector equation of a line in 2d and 3d - IB Questionbank | RevisionDojo

Practice AHL 3.11—Vector equation of a line in 2d and 3d with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

A line L has the vector equation

\vec{r}=\left[\begin{array}{c} 1 \\ 2 \\ -3 \end{array}\right]+\lambda\left[\begin{array}{c} 2 \\ -1 \\ 4 \end{array}\right]

Find the coordinates of the point on line L that is closest to the point $\mathrm{A}=(3$ , $-2,1)$ .

[6]

Question 2

HLPaper 1

A line L is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

Find the value of $\lambda$ forwhichthepointonthelineliesontheplanex $+y+z=5$ .

[4]

Question 3

HLPaper 1

A line passes through the point $P(4,-1,2)$ and is parallel to the vector $\left[\begin{array}{c}2 \\ 0 \\ -3\end{array}\right]$ .

Write down the vector equation of the line.

[3]

Question 4

HLPaper 1

The line $\mathrm{L}_{1}$ isgivenby $\vec{r}=\left[\begin{array}{l}1 \\ 3 \\ 2\end{array}\right]+\lambda\left[\begin{array}{c}2 \\ -1 \\ 1\end{array}\right]$ and point $\mathrm{A}=(5,1,3)$ .

Find the shortest distance from point A to line $\mathrm{L}_{1}$ .

[5]

Question 5

HLPaper 1

A line in 3D passes through the points $P(0,1,-2)$ and $Q(2,3,0)$ .

Find its vector equation in parametric form.

[4]

Question 6

HLPaper 1

Determine whether the point

P=\left[\begin{array}{l} 5 \\ 0 \\ 7 \end{array}\right]

lies on the line

\vec{r}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]+\lambda\left[\begin{array}{c} 2 \\ -2 \\ 2 \end{array}\right]

Determine whether the point

P=\left[\begin{array}{l} 5 \\ 0 \\ 7 \end{array}\right]

lies on the line

\vec{r}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]+\lambda\left[\begin{array}{c} 2 \\ -2 \\ 2 \end{array}\right]

[4]

Question 7

HLPaper 1

Two points $A(1,2)$ and $B(5,6)$ lie on a line in 2 D .

Find the vector equation of the line passing through both points.

[4]

Question 8

HLPaper 1

Find the point of intersection between the lines

\left[\begin{array}{l} 2+\lambda \\ 3-\lambda \end{array}\right]=\left[\begin{array}{c} 4-2 \mu \\ -1+2 \mu \end{array}\right]

[4]

Question 9

HLPaper 1

Two lines intersect at a point. Line 1 has equation

\vec{r}_{1}=\left[\begin{array}{c} 1 \\ 0 \\ -1 \end{array}\right]+\lambda\left[\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right]

Line 2 has equation

\vec{r}_{2}=\left[\begin{array}{c} 3 \\ -2 \\ 2 \end{array}\right]+\mu\left[\begin{array}{c} -1 \\ 3 \\ -2 \end{array}\right]

Find the point of intersection.

[5]

Question 10

HLPaper 1

A line in 3D has vector equation

\vec{r}=\left[\begin{array}{l} 1 \\ 4 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 3 \\ -1 \\ -2 \end{array}\right]

Find the value of $\lambda$ such that the corresponding point lies in the $x y$ -plane.

[4]

AHL 3.11—Vector equation of a line in 2d and 3d Questionbank