Introduction to vectors

Represent quantities as vectors in component and magnitude–direction form; perform vector arithmetic; apply dot products, projections, and geometric proofs with vectors; model projectile motion using vector components.

Vectors · ME-12-03

Sub-topics

Vector concepts

Represent vectors in component form using

\mathbf{i}

and

\mathbf{j}

; find magnitude and direction; perform vector addition, subtraction and scalar multiplication; find unit vectors.

Dot product and geometry

Calculate the scalar (dot) product of two vectors; find the angle between vectors; test for perpendicularity and parallelism; find scalar and vector projections; prove geometric results using vectors.

Projectile motion

Model projectile motion using horizontal and vertical vector components; derive and apply equations for time of flight, range, and maximum height; interpret results in real-world contexts.

Open Math

Introduction to vectors

Vectors · ME-12-03

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Vector concepts

Q1Straightforward

Find the magnitude of the vector

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

Q2Straightforward

The vector

\mathbf{u} = 6\mathbf{i} + 8\mathbf{j}

. Find the

x

-component of the unit vector

\hat{\mathbf{u}}

in the direction of

\mathbf{u}

Q3Straightforward

Given

\mathbf{v} = 3\mathbf{i} - 2\mathbf{j}

, find the vector

4\mathbf{v}

. Express your answer as a coordinate pair

(x, y)

Q4Straightforward

Vectors

\mathbf{a} = (2, 5)

and

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

. Find

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

to 2 decimal places.

Q5Moderate

Given

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

and

\mathbf{w} = \mathbf{i} - 4\mathbf{j}

, find

|3\mathbf{v} - 2\mathbf{w}|

to 2 decimal places.

Q6Moderate

A vector has magnitude 10 and makes an angle of 30° with the positive

x

-axis. Find its

x

-component to 2 decimal places.

Q7Moderate

Find the angle (in degrees) that the vector

\mathbf{v} = -\sqrt{3}\,\mathbf{i} + \mathbf{j}

makes with the positive

x

-axis. Give an exact answer.

Q8Moderate

Given

\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}

and

\mathbf{b} = 3\mathbf{i} - \mathbf{j}

, find

\mathbf{a} + \mathbf{b}

. Express your answer as a coordinate pair

(x, y)

Q9Moderate

Three vectors satisfy

\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}

. Given

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

and

\mathbf{b} = (-5, 1)

, find

|\mathbf{c}|

to 4 decimal places.

Q10Challenging

Two forces act on an object:

\mathbf{F}_1 = 4\mathbf{i} + 6\mathbf{j}

N and

\mathbf{F}_2 = 3\mathbf{i} - 2\mathbf{j}

N. Find the angle (to the nearest degree) that the resultant force makes with the positive

x

-axis.

Q11Challenging

A boat travels at 5 km/h in the direction of the vector

3\mathbf{i} + 4\mathbf{j}

. A current flows at 2 km/h in the direction

-\mathbf{i}

. Find the magnitude of the actual velocity of the boat to 4 decimal places.

Q12Challenging

For

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

, find the positive value of

\lambda

such that

|\lambda\mathbf{v}| = 10

. Give your answer to 4 decimal places.

Dot product and geometry

Q13Straightforward

Find the dot product

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

where

\mathbf{a} = (3, 4)

and

\mathbf{b} = (2, 1)

Q14Straightforward

Find

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

where

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

and

\mathbf{b} = (5, 4)

Q15Straightforward

Find

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

where

\mathbf{a} = (3, -4)

and

\mathbf{b} = (4, 3)

Q16Straightforward

Find the angle between

\mathbf{a} = (1, 0)

and

\mathbf{b} = (1, 1)

. Give your answer in degrees.

Q17Moderate

Find the angle between

\mathbf{a} = (3, 4)

and

\mathbf{b} = (4, 3)

to 2 decimal places. Give your answer in degrees.

Q18Moderate

Find the positive value of

k

such that the vectors

(k,\, k^2)

and

(3,\, -1)

are perpendicular.

Q19Moderate

Find the scalar projection of

\mathbf{a} = (5, 0)

onto

\mathbf{b} = (3, 4)

Q20Moderate

Find the vector projection of

\mathbf{a} = (7, 1)

onto

\mathbf{b} = (2, 1)

. Give your answer as a coordinate pair

(x, y)

Q21Moderate

Find the value of

k

such that the vectors

(k + 1,\, k)

and

(2,\, -3)

are perpendicular.

Q22Challenging

Points

A

B

C

have position vectors

\mathbf{a} = (2, 3)

\mathbf{b} = (5, 7)

\mathbf{c} = (4, 1)

. Find the angle

\angle ABC

(at vertex

B

) to 2 decimal places. Give your answer in degrees.

Q23Challenging

A force

\mathbf{F}

has magnitude 13 N and acts in the direction of the vector

(5, 12)

. An object is displaced by

\mathbf{d} = (4, 1)

m. The work done is

W = \mathbf{F} \cdot \mathbf{d}

(in joules). Find

W

Q24Challenging

Let

\mathbf{a}

and

\mathbf{b}

be vectors with

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

. Prove that

(\mathbf{a} + \mathbf{b})

and

(\mathbf{a} - \mathbf{b})

are perpendicular.

Projectile motion

Q25Straightforward

A ball is projected at 20 m/s at an angle of 30° above the horizontal. Taking

g = 10

m/s², find the time of flight in seconds (the time until it returns to the same height).

Q26Straightforward

A projectile is launched at 30 m/s at 45° above the horizontal. Taking

g = 10

m/s², find the horizontal range in metres.

Q27Straightforward

A projectile is launched at 20 m/s at 60° above the horizontal. Taking

g = 10

m/s², find the maximum height reached in metres.

Q28Straightforward

A projectile is launched at 25 m/s at an angle where

\sin\alpha = \tfrac{3}{5}

and

\cos\alpha = \tfrac{4}{5}

. Find the initial horizontal velocity component in m/s.

Q29Moderate

A ball is projected at 25 m/s with

\sin\alpha = \tfrac{3}{5}

\cos\alpha = \tfrac{4}{5}

. Taking

g = 10

m/s², find the vertical velocity in m/s at

t = 1

Q30Moderate

A ball is projected at 25 m/s with

\sin\alpha = \tfrac{3}{5}

\cos\alpha = \tfrac{4}{5}

(same ball as Q5). Taking

g = 10

m/s², find the height of the ball in metres at

t = 2

Q31Moderate

A projectile is launched at 20 m/s at 60° above the horizontal. Taking

g = 10

m/s², find the time to reach maximum height. Give your answer to 2 decimal places.

Q32Moderate

A projectile is launched at 25 m/s with

\sin\alpha = \tfrac{4}{5}

\cos\alpha = \tfrac{3}{5}

. Taking

g = 10

m/s², find the maximum height in metres.

Q33Moderate

A projectile is launched at 30 m/s at 30° above the horizontal. Taking

g = 10

m/s², find the horizontal range in metres to 2 decimal places.

Q34Challenging

A ball is projected at 20 m/s at 30° above the horizontal. Taking

g = 10

m/s², find the speed of the ball in m/s at

t = 1.5

s. Give your answer to 2 decimal places.

Q35Challenging

A ball is kicked at 45° and lands 80 m away on the same level. Taking

g = 10

m/s², find the initial speed

u

to 2 decimal places.

Q36Challenging

A projectile is launched at 30 m/s with

\sin\alpha = \tfrac{4}{5}

\cos\alpha = \tfrac{3}{5}

. Taking

g = 10

m/s², find the horizontal range in metres.

Worked solutions and answers at openmath.au/year-12/extension-1/introduction-to-vectors