Year 12 Maths Extension 1 (2027) Vectors

2D vectors in component form

Q: How do you write a vector in component form?

As x i plus y j, the ordered pair x comma y, or a column vector.

Q: How do you find the magnitude?

The magnitude of x i plus y j is the square root of x squared plus y squared.

Q: How do you find a unit vector?

Divide the vector by its magnitude, a-hat equals a over the magnitude of a.

Q: How do you find the vector between two points?

Subtract the tail's coordinates from the head's.

Q: How do you make a vector of a given length?

Multiply the unit vector by the required magnitude.

20 practice questions 2 video lessons Theory + worked examples

Work with 2D vectors in component form for NSW Year 12 Mathematics Extension 1. Any vector in the plane can be written using the i and j unit vectors, as an ordered pair, or as a column vector, linking geometry to coordinates.

You will learn to form the vector between two points, find its magnitude and midpoint, and use the zero and unit vectors — the component form that makes vector calculations quick and exact throughout the Extension 1 course.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

Vectors - Vectors in component form (by Class Mathematics) Watch
Vectors : magnitude of a vector in 2D : ExamSolutions Watch

Theory

In component form a 2D vector is $x i + y j$ , with magnitude $\sqrt{x^{2} + y^{2}}$ and unit vector $\hat{a} = \frac{1}{| a |} a$ . This NSW Year 12 Mathematics Extension 1 topic is NESA outcome ME1-12-02.

In component form a 2D vector is written $x i + y j$ , as an ordered pair $(x, y)$ , or as a column vector $(\begin{matrix} x \\ y \end{matrix})$ , where $i$ and $j$ are the perpendicular unit vectors along the axes.

From $A (x, y)$ to $B (u, v)$ : $\vec{A B} = (u - x) i + (v - y) j$ . The magnitude is $| x i + y j | = \sqrt{x^{2} + y^{2}}$ , the hypotenuse of the component right triangle.

The unit vector $\hat{a} = \frac{1}{| a |} a$ has magnitude $1$ in the direction of $a$ .

NESA link. Part of the Year 12 Introduction to vectors focus area, outcome ME1-12-02 ("operates with 2D and 3D vectors and uses 2D vectors to solve problems involving motion in two dimensions") with MAO-WM-01.

a = x i + y j

has magnitude

\sqrt{x^{2} + y^{2}}

\hat{a} = \frac{1}{| a |} a

has length

1

along

a

| x i + y j | = \sqrt{x^{2} + y^{2}}, \hat{a} = \frac{1}{| a |} a, \vec{A B} = (u - x) i + (v - y) j .

magnitude = sqrt(x^2 + y^2); unit vector = a / |a|; AB = (u-x)i + (v-y)j

Scaling to a length. A vector parallel to $a$ with magnitude $m$ is $m \hat{a}$ — the unit vector scaled to the required length.

How to work in components

Magnitude: $| x i + y j | = \sqrt{x^{2} + y^{2}}$ .
Vector between points: subtract the tail's coordinates from the head's.
Unit vector: divide each component by the magnitude.
Set a length: multiply the unit vector by the required magnitude $m$ .

Example 1 — Magnitude

Find the magnitude of

a = 5 i - 12 j

Solution

| a |

=

\sqrt{5^{2} + (- 12)^{2}} = \sqrt{169} = 13

|a| = 13

$| a | = 13$ .

Example 2 — Vector between points

Given

A (1, - 3)

and

B (9, 3)

, find

\vec{A B}

and

| \vec{A B} |

Solution

$\vec{A B}$	$=$	$(9 - 1) i + (3 - (- 3)) j = 8 i + 6 j$
$\| \vec{A B} \|$	$=$	$\sqrt{8^{2} + 6^{2}} = 10$

AB = 8i + 6j, |AB| = 10

$\vec{A B} = 8 i + 6 j$ , $| \vec{A B} | = 10$ .

Example 3 — Unit vector

Find the unit vector in the direction of

a = 9 i - 12 j

Solution

$| a | = \sqrt{81 + 144} = 15$ .

\hat{a}

=

\frac{9 i - 12 j}{15} = 0.6 i - 0.8 j

a-hat = 0.6i - 0.8j

$\hat{a} = 0.6 i - 0.8 j$ .

Example 4 — Scale to a length

Find the vector parallel to

3 i + 4 j

with magnitude

20

Solution

The unit vector is $\frac{3 i + 4 j}{5}$ .

20 \hat{a}

=

4 (3 i + 4 j) = 12 i + 16 j

12i + 16j

The vector is $12 i + 16 j$ .

Common pitfalls

Head minus tail.

\vec{A B}

subtracts

A

's coordinates from

B

's, keeping the sign of negatives.

Unit vector length. A unit vector must have magnitude

1

— divide each component by

| a |

Dividing by $| a |$ once. Both components share the same denominator

| a |

Setting a length. A vector parallel to

a

with magnitude

m

m \hat{a}

, not

m a

Frequently asked questions

How do you write a vector in component form?

As $x i + y j$ , the ordered pair $(x, y)$ , or a column vector.

How do you find the magnitude?

$| x i + y j | = \sqrt{x^{2} + y^{2}}$ .

How do you find a unit vector?

Divide the vector by its magnitude: $\hat{a} = \frac{1}{| a |} a$ .

How do you find the vector between two points?

Subtract the tail's coordinates from the head's: $\vec{A B} = (u - x) i + (v - y) j$ .

How do you make a vector of a given length?

Multiply the unit vector by the required magnitude $m$ .

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Operating with vectors in two dimensions

2D vectors in component form

📖 Theory

How to work in components

Common pitfalls

Frequently asked questions

Theory