Three-dimensional vectors

Represent vectors in 3D using components

\mathbf{i}

\mathbf{j}

\mathbf{k}

; find magnitudes and unit vectors; compute the dot product; find angles between vectors in 3D; determine perpendicularity and parallelism.

Worked examples

Magnitude, dot product and perpendicularity

Straightforward

Problem

Given

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

and

\mathbf{b} = (2,-1,0)

, find

|\mathbf{a}|

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

, and determine whether they are perpendicular.

Angle between two vectors

Moderate

Problem

Find the angle between

\mathbf{p} = (1,0,1)

and

\mathbf{q} = (1,1,0)

Finding an unknown component

Challenging

Problem

Find the value of

t

such that

\mathbf{u} = (t,1,2)

and

\mathbf{v} = (3,t,-1)

are perpendicular.

Practise

Q1·Straightforward

Find

|\mathbf{u}|

where

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

Q2·Straightforward

Find

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

where

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

and

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

Q3·Straightforward

Let

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

and

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

. Find the

\mathbf{k}

-component of

\mathbf{a}+\mathbf{b}

Q4·Straightforward

Find

|\mathbf{v}|

where

\mathbf{v} = -6\mathbf{i}+8\mathbf{k}

Q5·Straightforward

Find

|\mathbf{w}|

where

\mathbf{w} = 5\mathbf{j}-12\mathbf{k}

Q6·Straightforward

Find

\mathbf{u}\cdot\mathbf{v}

where

\mathbf{u} = \mathbf{i}-\mathbf{k}

and

\mathbf{v} = 2\mathbf{i}+3\mathbf{j}+4\mathbf{k}

Q7·Moderate

Find the angle, in degrees, between

\mathbf{a} = \mathbf{i}+\mathbf{j}

and

\mathbf{b} = \mathbf{i}+\mathbf{k}

Q8·Moderate

Find the value of

m

such that

\mathbf{u} = m\mathbf{i}-\mathbf{j}+3\mathbf{k}

is perpendicular to

\mathbf{v} = 2\mathbf{i}+m\mathbf{j}+\mathbf{k}

Q9·Moderate

Find the angle, in degrees, between

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

and

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

. Round to two decimal places.

Q10·Moderate

Find the

x

-component of the unit vector in the direction of

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

. Give your answer as a fraction.

Q11·Moderate

Which statement correctly describes

\mathbf{a} = (2,-4,6)

and

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

They are parallel, because

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

They are perpendicular, because

\mathbf{a}\cdot\mathbf{b} = 0

They are neither parallel nor perpendicularThey are parallel, because

|\mathbf{a}| = 2|\mathbf{b}|

Q12·Moderate

Let

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

and

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

. Find

|\mathbf{a}+\mathbf{b}|

, rounding to two decimal places.

Q13·Challenging

Find the value of

t

such that

(t,3,-2)

and

(1,t,4)

are perpendicular.

Q14·Challenging

Find the angle, in degrees, that the main diagonal of a unit cube makes with one of its edges. Take the diagonal from

(0,0,0)

(1,1,1)

and the edge along the positive

x

-axis. Round to two decimal places.

Q15·Challenging

Vectors

\mathbf{a}

and

\mathbf{b}

satisfy

|\mathbf{a}| = 3

|\mathbf{b}| = 4

and

\mathbf{a}\cdot\mathbf{b} = 6

. Find the angle between them in degrees.

Open Math

Three-dimensional vectors

Vectors · MEX-12-05

Name:

Date:

Q1Straightforward

Find

|\mathbf{u}|

where

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

Q2Straightforward

Find

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

where

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

and

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

Q3Straightforward

Let

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

and

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

. Find the

\mathbf{k}

-component of

\mathbf{a}+\mathbf{b}

Q4Straightforward

Find

|\mathbf{v}|

where

\mathbf{v} = -6\mathbf{i}+8\mathbf{k}

Q5Straightforward

Find

|\mathbf{w}|

where

\mathbf{w} = 5\mathbf{j}-12\mathbf{k}

Q6Straightforward

Find

\mathbf{u}\cdot\mathbf{v}

where

\mathbf{u} = \mathbf{i}-\mathbf{k}

and

\mathbf{v} = 2\mathbf{i}+3\mathbf{j}+4\mathbf{k}

Q7Moderate

Find the angle, in degrees, between

\mathbf{a} = \mathbf{i}+\mathbf{j}

and

\mathbf{b} = \mathbf{i}+\mathbf{k}

Q8Moderate

Find the value of

m

such that

\mathbf{u} = m\mathbf{i}-\mathbf{j}+3\mathbf{k}

is perpendicular to

\mathbf{v} = 2\mathbf{i}+m\mathbf{j}+\mathbf{k}

Q9Moderate

Find the angle, in degrees, between

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

and

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

. Round to two decimal places.

Q10Moderate

Find the

x

-component of the unit vector in the direction of

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

. Give your answer as a fraction.

Q11Moderate

Which statement correctly describes

\mathbf{a} = (2,-4,6)

and

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

A.They are parallel, because $\mathbf{a} = -2\mathbf{b}$
B.They are perpendicular, because $\mathbf{a}\cdot\mathbf{b} = 0$
C.They are neither parallel nor perpendicular
D.They are parallel, because $|\mathbf{a}| = 2|\mathbf{b}|$

Q12Moderate

Let

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

and

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

. Find

|\mathbf{a}+\mathbf{b}|

, rounding to two decimal places.

Q13Challenging

Find the value of

t

such that

(t,3,-2)

and

(1,t,4)

are perpendicular.

Q14Challenging

Find the angle, in degrees, that the main diagonal of a unit cube makes with one of its edges. Take the diagonal from

(0,0,0)

(1,1,1)

and the edge along the positive

x

-axis. Round to two decimal places.

Q15Challenging

Vectors

\mathbf{a}

and

\mathbf{b}

satisfy

|\mathbf{a}| = 3

|\mathbf{b}| = 4

and

\mathbf{a}\cdot\mathbf{b} = 6

. Find the angle between them in degrees.

Worked solutions and answers at openmath.au/year-12/extension-2/further-work-with-vectors/three-dimensional-vectors