Higher Maths · Worksheet 4

Vectors Worksheet | Higher Maths

Skills covered: Two- and three-dimensional vectors; position vectors; magnitude; scalar product; collinearity; internal division; three-dimensional angles.

18 questions62 marks3 graded sectionsFree PDF

Download PDF Vectors revision notes

Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.

These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if vectors still feels shaky; if you are confident, go straight to Section C and time yourself.

Section A

Prerequisite core skills

Revise earlier learning

Question 1

[3 marks]

Given

\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}

and

\mathbf{b} = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}

, express

2\mathbf{a} - \mathbf{b}

in unit-vector form.

Question 2

[2 marks]

Find the magnitude of

\mathbf{v} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}

.

Question 3

[3 marks]

Given

A(1, -2, 3)

and

B(5, 1, -1)

, find

\overrightarrow{AB}

and

|\overrightarrow{AB}|

.

Question 4

[2 marks]

Find

\mathbf{a} \cdot \mathbf{b}

for

\mathbf{a} = (3, -2, 1)

and

\mathbf{b} = (2, 4, -1)

.

Question 5

[3 marks]

Find

k

if

(2, k, -1)

is perpendicular to

(3, 2, 4)

.

Question 6

[2 marks]

The position vectors of

P

and

Q

are

2\mathbf{i} + \mathbf{j} - 3\mathbf{k}

and

-\mathbf{i} + 5\mathbf{j} + \mathbf{k}

. Find

\overrightarrow{PQ}

.

Section B

Topic practice

Build individual techniques

Question 7

[4 marks]

Show that the points

A(1, 2, 0)

,

B(3, 6, 2)

and

C(6, 12, 5)

are collinear.

Question 8

[4 marks]

The points

A(2, -1, 4)

and

B(8, 5, -2)

are joined by a line segment. Find the coordinates of the point

P

which divides

AB

internally in the ratio

AP : PB = 1 : 2

.

Question 9

[3 marks]

Find the angle between

\mathbf{a} = (2, 1, 2)

and

\mathbf{b} = (1, 2, -2)

.

Question 10

[4 marks]

Given

\mathbf{p} = (1, 2, 2)

and

\mathbf{q} = (2, -1, 2)

, calculate the acute angle between the vectors, to one decimal place.

Question 11

[4 marks]

In a triangle,

A(1, 0, 2)

,

B(5, 2, 1)

and

C(2, 4, 3)

. Find the coordinates of the midpoint

M

of

BC

and

\overrightarrow{AM}

.

Question 12

[3 marks]

Let

\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}

and

\mathbf{b} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k}

. Determine whether the vectors are perpendicular.

Question 13

[3 marks]

For

A(1, 3, -2)

,

B(4, 1, 0)

and

C(6, 5, 1)

, find

\overrightarrow{AB} \cdot \overrightarrow{AC}

.

Question 14

[3 marks]

Write

\mathbf{u} = (3, -4, 12)

as a unit vector in the direction of

\mathbf{u}

.

Section C

Exam-style and challenge

Combine skills, exam-style

Question 15

[5 marks]

For

A(0, 1, 2)

,

B(4, 3, 0)

and

C(1, 5, 4)

, calculate

\angle BAC

to one decimal place.

Question 16

[4 marks]

The points

A(2, 1, -1)

,

B(8, 7, 5)

, and

P(t, 4, 2)

lie on the same straight line. Find

t

.

Question 17

[5 marks]

The points

O = (0, 0, 0)

,

A = (4, 0, 0)

,

B = (0, 3, 0)

, and

P = (2, 1, 6)

are vertices of a tetrahedron. Find

\angle APB

to one decimal place.

Question 18

[5 marks]

A point

R

divides the line segment from

P(-2, 4, 1)

to

Q(7, -2, 10)

in the ratio

PR : RQ = 2 : 1

. Find

R

. Hence calculate

|\overrightarrow{PR}|

.

Looking for the answers?

We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.

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Frequently asked questions

Are the answers to these worksheets available online?

No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.

Are these official SQA or Qualifications Scotland papers?

No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.

How should I use the three sections?

Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.

Can I print the worksheet?

Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.

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