Vector language and operations | CAIE 9709 Pure Mathematics 3 Notes

Chapters

Pure Mathematics 3

Mathematics9709

Read the notation before calculating

One vector, three forms

arrow

↗ v

column

(2, −1, 3)

basis form

2i − j + 3k

2v: same direction, twice the length
−v: same length, opposite direction

The arrow, column and i, j, k forms describe the same vector; a scalar changes its length and possibly its direction.

$\mathbf v$ is a vector. In print it is bold; in handwritten work, underline the letter so it cannot be mistaken for a scalar.
$\mathbf i$ , $\mathbf j$ and $\mathbf k$ are unit vectors, so each has length 1. They point in the positive $x$ -, $y$ - and $z$ -directions.
The components in $\begin{pmatrix}2\\-1\\3\end{pmatrix}$ mean move 2 in $x$ , −1 in $y$ and 3 in $z$ .

\mathbf v=\begin{pmatrix}2\\-1\\3\end{pmatrix}

\mathbf v=2\mathbf i-\mathbf j+3\mathbf k

Key idea

A point

A=(2,-1,3)

is a location. A vector with the same three components is a directed movement. The numbers match, but the meanings are different.

Add, subtract and multiply components

Add or subtract matching components.
Multiplying by 3 keeps the direction and triples the magnitude. Multiplying by −3 also triples the magnitude, but reverses the direction.
Two non-zero vectors are parallel when one is a scalar multiple of the other.

Worked example

Combine two vectors

\mathbf u=(2,-1,3)

\mathbf v=(-1,4,0)

2\mathbf u=(4,-2,6)

Answer

2\mathbf u-\mathbf v=(5,-6,6)

(Pure Mathematics 2 & 3 Coursebook, Ch. 9.1–9.2)

Use the scale factor as a direction check

Your turnIndependent check [4 marks]

Let $\mathbf a=(2,-1,3)$ , $\mathbf b=(-1,4,0)$ and $\mathbf c=(-6,3,-9)$ . Find $2\mathbf a-\mathbf b$ , then describe the relationship between $\mathbf a$ and $\mathbf c$ .

Show worked answer

Answer

2\mathbf a-\mathbf b=(5,-6,6)

\mathbf c=-3\mathbf a

The vectors are parallel and point in opposite directions.

Read the notation before calculating

One vector, three forms

arrow

↗ v

column

(2, −1, 3)

basis form

2i − j + 3k

2v: same direction, twice the length
−v: same length, opposite direction

The arrow, column and i, j, k forms describe the same vector; a scalar changes its length and possibly its direction.

$\mathbf v$ is a vector. In print it is bold; in handwritten work, underline the letter so it cannot be mistaken for a scalar.
$\mathbf i$ , $\mathbf j$ and $\mathbf k$ are unit vectors, so each has length 1. They point in the positive $x$ -, $y$ - and $z$ -directions.
The components in $\begin{pmatrix}2\\-1\\3\end{pmatrix}$ mean move 2 in $x$ , −1 in $y$ and 3 in $z$ .

\mathbf v=\begin{pmatrix}2\\-1\\3\end{pmatrix}

\mathbf v=2\mathbf i-\mathbf j+3\mathbf k

Key idea

A point

A=(2,-1,3)

is a location. A vector with the same three components is a directed movement. The numbers match, but the meanings are different.

Add, subtract and multiply components

Add or subtract matching components.
Multiplying by 3 keeps the direction and triples the magnitude. Multiplying by −3 also triples the magnitude, but reverses the direction.
Two non-zero vectors are parallel when one is a scalar multiple of the other.

Worked example

Combine two vectors

\mathbf u=(2,-1,3)

\mathbf v=(-1,4,0)

2\mathbf u=(4,-2,6)

Answer

2\mathbf u-\mathbf v=(5,-6,6)

(Pure Mathematics 2 & 3 Coursebook, Ch. 9.1–9.2)

Use the scale factor as a direction check

Your turnIndependent check [4 marks]

Let $\mathbf a=(2,-1,3)$ , $\mathbf b=(-1,4,0)$ and $\mathbf c=(-6,3,-9)$ . Find $2\mathbf a-\mathbf b$ , then describe the relationship between $\mathbf a$ and $\mathbf c$ .

Show worked answer

Answer

2\mathbf a-\mathbf b=(5,-6,6)

\mathbf c=-3\mathbf a

The vectors are parallel and point in opposite directions.