Topic 3.16 · HL only

Vector equations in 2&3D (+ angle between two lines): notes and practice questions

Q: What does Vector equations in 2&3D (+ angle between two lines) cover in IB Maths AI?

Vector Equation of a Line: r = a + λ b. r: position vector of any point on the line. a: position vector of a known point on the line.

Q: Is Vector equations in 2&3D (+ angle between two lines) SL or HL?

Vector equations in 2&3D (+ angle between two lines) is HL only. SL students are not examined on it.

Q: How do I revise Vector equations in 2&3D (+ angle between two lines) for IB Maths AI?

Start from the core idea: vector Equation of a Line: r = a + λ b. In the exam: the scalar product is asked far more often than the vector product, usually as "find the angle between". The cross product turns up for areas and for a normal direction. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Vector equations in 2&3D (+ angle between two lines)?

FourtyFive has 7 Vector equations in 2&3D (+ angle between two lines) questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Vector equations in 2&3D (+ angle between two lines) practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Vector equations in 2&3D (+ angle between two lines) answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

6 min read Jump to 7 practice questions

Summary

Vector Equation of a Line: $r = a + \lambda b$
$r$ : position vector of any point on the line.
$a$ : position vector of a known point on the line.
$b$ : direction vector parallel to the line.
$\lambda$ : scalar parameter.
Finding Direction Vector (from two points A and B): $b = \vec{AB} = \vec{OB} - \vec{OA}$
**Checking if a Point $C$ Lies on a Line:**
Equate point's position vector $c$ to the line equation:

$\begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \lambda \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$

Solve for $\lambda$ from each component equation.
Point lies on the line if a single, consistent $\lambda$ value satisfies all components.
Angle Between Two Lines:
Determined by their direction vectors $b_1$ and $b_2$ .
Acute angle formula: $\cos \theta = \frac{|b_1 \cdot b_2|}{|b_1||b_2|}$
Absolute value of dot product ensures the acute angle.
**Shortest Distance from a Point $A$ to a Line:**
Define a general point $B$ on the line using parameter $\lambda$ .
Form the displacement vector $\vec{AB} = \vec{OB} - \vec{OA}$ .
Apply perpendicular condition: $\vec{AB} \cdot b = 0$ (where $b$ is the line's direction vector).
Solve for $\lambda$ .
Substitute $\lambda$ back into $\vec{AB}$ and calculate its magnitude $|\vec{AB}|$ .
GDC Tips:
Use GDC for solving systems of linear equations (e.g., when checking if a point lies on a line).
Use GDC's vector tools for dot product and magnitude calculations.

How it is examined

The scalar product is asked far more often than the vector product, usually as "find the angle between". The cross product turns up for areas and for a normal direction. Proofs of the general properties are explicitly not required, so a question asking a student to prove distributivity is out of syllabus. The component formulas at the end are what physics-adjacent Paper 3 questions lean on.

Given in the booklet

Both products in component and in magnitude-angle form, and the area of a parallelogram.

Key ideas

The definition and calculation of the scalar product of two vectors.
The angle between two vectors, and the acute angle between two lines.
The definition and calculation of the vector product of two vectors.
The geometric interpretation of $|\boldsymbol{v} \times \boldsymbol{w}|$ .

Not assessed

Not required: generalized properties and proofs of scalar and cross product.

Linking questions

Other contexts, computer graphics: lighting, normalising one vector onto another to determine the intensity of light on a surface. Perspective, projecting a three-dimensional vector onto a two-dimensional plane using the scalar product.
Other contexts, physics: torque, where the magnitude of the rotational force applied to a point is the magnitude of the cross product of the length of the lever and the force applied to it, and the direction of the torque says whether the force tightens or loosens the bolt. Electromagnetic forces and the right and left hand rules, $W = F \cdot d$ . Forces, and what component of one force acts in the direction of another, which matters for strain analysis.
Links to other subjects: magnetic forces and fields, and dynamics (physics).
TOK: what counts as understanding in mathematics? Is it more than just getting the right answer?

Practice questions

7 questions · 6 medium · 1 hard

Showing 7 of 7

Question 1

MediumPaper 1 · calculator7 marks

(a)

The paths of two automated guided vehicles (AGVs), $L_1$ and $L_2$ , in a large warehouse are modelled by the following vector equations, where $k\in\mathbb{R}$ is a constant:

$L_1:\boldsymbol{r}=\begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} k \ 5 \ -6 \end{pmatrix}$

$L_2: \boldsymbol{r} = \begin{pmatrix} 10 \ -1 \ 8 \end{pmatrix} + \mu \begin{pmatrix} k+2 \ 3 \ 5 \end{pmatrix}$

It is known that the paths of the AGVs are perpendicular.

(a) Find the possible value(s) for $k$ .

[3]

(b)

(b) In the case that $k < 0$ , determine whether the lines intersect.

[4]

Question 2

HardPaper 2 · calculator19 marks

(a)

(a) A space probe, 'Voyager Alpha', is launched from a space station. The position of the probe at time $t$ days after launch is given by the vector

$\mathbf{r} = \begin{pmatrix} 10 \\ 20 \\ 5 \end{pmatrix} + t \begin{pmatrix} 80 \\ 40 \\ 10 \end{pmatrix}$

Distances are measured in thousands of kilometres.

Find the position vector of the probe $3$ days after launch.

[3]

(b)

(b) A second space probe, 'Explorer Beta', is launched from a different station. The position vector of this probe is given by the vector

$\mathbf{s} = \begin{pmatrix} -50 \\ -30 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 70 \\ 50 \\ 12 \end{pmatrix}$

Determine if the two flight paths intersect and, if so, state the point of intersection.

[5]

(c)

State, with a reason, whether the two probes actually collide.

[2]

(d)

(d) Calculate the distance between the two space stations (the initial launch points).

[3]

(e)

(e) Calculate the shortest distance that ever exists between the two probes and the time when this occurs. Assume $t \ge 0$ days.

[6]

Question 3

MediumPaper 1 · calculator9 marks

(a)

A triangular section of a modern architectural roof is being designed. The vertices of this section are represented by points P, Q, and R in a 3D coordinate system, where distances are measured in metres.

Point P is at $(0, 0, 2)$ .

Point Q is at $(4, 7, 3)$ .

Point R is at $(9, 2, 2)$ .

Calculate the vector product $\vec{PQ} \times \vec{PR}$ .

[2]

(b)

Hence, find the area of the triangular roof section.

[2]

(c)

A support beam is to be installed from point Q perpendicular to the edge PR. Find the length of this support beam.

[3]

(d)

The building's base lies on the horizontal plane (z=0). Find the acute angle that the roof section makes with the horizontal plane.

[2]

Question 4

MediumPaper 1 · calculator7 marks

(a)

(a) A drone is programmed to fly along a straight path $L$ . Its position at time $t$ (in minutes) can be modelled by the vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ , where the coordinates are in metres. A stationary sensor $S$ is located at the point $(10, 4, 4)$ .

Find the coordinates of the point $P$ on the drone's path that is closest to the sensor $S$ .

[4]

(b)

(b) Find a vector that is perpendicular to both the drone's path $L$ and the line segment connecting point $P$ to the sensor $S$ .

[3]

Question 5

MediumPaper 1 · calculator5 marks

A force $\vec{F}$ acts on a particle. The effect of this force is often analyzed by decomposing it into components parallel and perpendicular to a given direction vector $\vec{d}$ (where $\vec{d} \neq \vec{0}$ ).

The scalar component of $\vec{F}$ in the direction of $\vec{d}$ is given by $F_{\text{parallel}} = \frac{\vec{F} \cdot \vec{d}}{|\vec{d}|}$ .

The magnitude of the component of $\vec{F}$ perpendicular to $\vec{d}$ is given by $F_{\text{perpendicular}} = \frac{|\vec{F} \times \vec{d}|}{|\vec{d}|}$ .

(a) Show that $|\vec{F}|^2 = F_{\text{parallel}}^2 + F_{\text{perpendicular}}^2$ .

Question 6

MediumPaper 1 · calculator7 marks

(a)

(a) A reconnaissance drone is launched from a base station at point $A(5, 2, 1)$ relative to an origin $O$ , where lengths are measured in metres and time, $t$ , is measured in seconds. It flies in a straight line with vector equation

$\vec{r} = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 4 \\ 4 \\ 2 \end{pmatrix}$

Find the speed of the reconnaissance drone (in m/s).

[2]

(b)

(b) A patrol drone is dispatched from a different location with position vector $\begin{pmatrix} -4 \\ 29 \\ 37 \end{pmatrix}$ , relative to $O$ . It flies with a constant velocity vector of $\begin{pmatrix} 7 \\ -5 \\ -10 \end{pmatrix}$ to intercept the reconnaissance drone.

Write down the vector equation for $\vec{p}$ , that models the flight of the patrol drone.

[1]

(c)

[4]

Question 7

MediumPaper 1 · calculator6 marks

(a)

A mechanic applies a force, $\boldsymbol{F} = \begin{pmatrix} 10 \\ k \\ -50 \end{pmatrix}$ N, to the end of a wrench. The direction of this force is perpendicular to a vector $\boldsymbol{d} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}$ .

(a) Find the value of $k$ .

[2]

(b)

The wrench applies the force at a point P, which has a position vector $\boldsymbol{r} = \begin{pmatrix} 0.2 \\ 0.1 \\ 0 \end{pmatrix}$ m relative to the bolt (which is at the origin). The torque, $\boldsymbol{\tau}$ , is given by the vector equation $\boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F}$ . The actual force applied is given by $\boldsymbol{F} = c \begin{pmatrix} 10 \\ 4 \\ -50 \end{pmatrix}$ , where $c \in \mathbb{R}^+$ .

(b) Given that the magnitude of the torque is 20 Nm, find the value of $c$ .

[4]

Every Vector equations in 2&3D (+ angle between two lines) question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Using your own wrong value after failing a "show that." All follow through is withdrawn for the rest of that question.

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Can't find what you're looking for? Email our student team.

What does Vector equations in 2&3D (+ angle between two lines) cover in IB Maths AI?

Vector Equation of a Line: r = a + λ b. r: position vector of any point on the line. a: position vector of a known point on the line.

Is Vector equations in 2&3D (+ angle between two lines) SL or HL?

Vector equations in 2&3D (+ angle between two lines) is HL only. SL students are not examined on it.

How do I revise Vector equations in 2&3D (+ angle between two lines) for IB Maths AI?

Start from the core idea: vector Equation of a Line: r = a + λ b. In the exam: the scalar product is asked far more often than the vector product, usually as "find the angle between". The cross product turns up for areas and for a normal direction. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Vector equations in 2&3D (+ angle between two lines)?

FourtyFive has 7 Vector equations in 2&3D (+ angle between two lines) questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Vector equations in 2&3D (+ angle between two lines) practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Vector equations in 2&3D (+ angle between two lines) answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Vector equations in 2&3D (+ angle between two lines): notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Vector equations in 2&3D (+ angle between two lines) question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Vector equations in 2&3D (+ angle between two lines) question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.