## Vector

 8.1 Vector

Example
 Scalar Quantity Vector Quantity Distance Displacement Speed Velocity Mass Weight

Vector
 Figure

 Characteristic
• A vector is usually represented by a directed line segment drawn as an arrow.
• The length of the line represents the magnitude or the size of the vector and the arrow indicates the direction of the vector.
• A vector from an initial point $$A$$ to a terminal point $$B$$ can be written as $$\overrightarrow{AB}$$$$\utilde{a}$$$$AB$$ or $$a$$.
• Vector $$-\overrightarrow{AB}$$ represents a vector in the opposite direction as that is $$\overrightarrow{AB}$$, such that $$\overrightarrow{BA}=-\overrightarrow{AB}$$.
• Two vectors are equal if and only if both the vectors have the same magnitude and direction.
• A zero vector $$\utilde{0}$$ has magnitude zero and its direction cannot be determined.

• The vector $$\utilde{a}$$ multiplied by the scalar $$k$$ is also a vector and is written as $$k\utilde{a}$$ where

(i) $$\begin{vmatrix} ka \end{vmatrix}=k \begin{vmatrix} ka \end{vmatrix}$$

(ii) if $$k \gt0$$, then $$k\utilde{a}$$ has the same direction as $$\utilde{a}$$.

(iii) if $$k \lt0$$, then $$k\utilde{a}$$ has the opposite direction as $$\utilde{a}$$.

• Vector $$\utilde{a}$$ and $$\utilde{b}$$ are parallel if and only if $$\utilde{a}=k\utilde{a}$$ where $$k$$ is a constant.

Example $$1$$
 Question

Given the vector $$\utilde{m}$$.

Express the vector below in term of $$\utilde{m}$$.

 Solution

Based on the given diagram,

the vector has twice the magnitude of the vector $$\utilde{m}$$ and also due in same direction with vector $$\utilde{m}$$.

Hence, the vector in term of $$\utilde{m}$$ is:

$$2\utilde{m}$$

Example $$2$$
 Question

Given a pair of vector.

Determine whether the pair of vector is parallel or not.

$$\overrightarrow{EF}=\dfrac{1}{3}\utilde{r}$$

$$\overrightarrow{FG}=9\utilde{r}$$

 Solution

Vector $$\utilde{a}$$ and $$\utilde{b}$$ are parallel if and only if $$\utilde{a}=k\utilde{b}$$ where $$k$$ is a constant.

$$\overrightarrow{EF}=\dfrac{1}{3}\utilde{r}$$,

Then,

$$\utilde{r}=3\overrightarrow{EF}$$.

Hence,

\begin{aligned} \overrightarrow{FG}&=9(3\overrightarrow{EF})\\\\ &=27\overrightarrow{EF}. \end{aligned}

$$\therefore \overrightarrow{EF}$$ and $$\overrightarrow{FG}$$ are parallel.

## Vector

 8.1 Vector

Example
 Scalar Quantity Vector Quantity Distance Displacement Speed Velocity Mass Weight

Vector
 Figure

 Characteristic
• A vector is usually represented by a directed line segment drawn as an arrow.
• The length of the line represents the magnitude or the size of the vector and the arrow indicates the direction of the vector.
• A vector from an initial point $$A$$ to a terminal point $$B$$ can be written as $$\overrightarrow{AB}$$$$\utilde{a}$$$$AB$$ or $$a$$.
• Vector $$-\overrightarrow{AB}$$ represents a vector in the opposite direction as that is $$\overrightarrow{AB}$$, such that $$\overrightarrow{BA}=-\overrightarrow{AB}$$.
• Two vectors are equal if and only if both the vectors have the same magnitude and direction.
• A zero vector $$\utilde{0}$$ has magnitude zero and its direction cannot be determined.

• The vector $$\utilde{a}$$ multiplied by the scalar $$k$$ is also a vector and is written as $$k\utilde{a}$$ where

(i) $$\begin{vmatrix} ka \end{vmatrix}=k \begin{vmatrix} ka \end{vmatrix}$$

(ii) if $$k \gt0$$, then $$k\utilde{a}$$ has the same direction as $$\utilde{a}$$.

(iii) if $$k \lt0$$, then $$k\utilde{a}$$ has the opposite direction as $$\utilde{a}$$.

• Vector $$\utilde{a}$$ and $$\utilde{b}$$ are parallel if and only if $$\utilde{a}=k\utilde{a}$$ where $$k$$ is a constant.

Example $$1$$
 Question

Given the vector $$\utilde{m}$$.

Express the vector below in term of $$\utilde{m}$$.

 Solution

Based on the given diagram,

the vector has twice the magnitude of the vector $$\utilde{m}$$ and also due in same direction with vector $$\utilde{m}$$.

Hence, the vector in term of $$\utilde{m}$$ is:

$$2\utilde{m}$$

Example $$2$$
 Question

Given a pair of vector.

Determine whether the pair of vector is parallel or not.

$$\overrightarrow{EF}=\dfrac{1}{3}\utilde{r}$$

$$\overrightarrow{FG}=9\utilde{r}$$

 Solution

Vector $$\utilde{a}$$ and $$\utilde{b}$$ are parallel if and only if $$\utilde{a}=k\utilde{b}$$ where $$k$$ is a constant.

$$\overrightarrow{EF}=\dfrac{1}{3}\utilde{r}$$,

Then,

$$\utilde{r}=3\overrightarrow{EF}$$.

Hence,

\begin{aligned} \overrightarrow{FG}&=9(3\overrightarrow{EF})\\\\ &=27\overrightarrow{EF}. \end{aligned}

$$\therefore \overrightarrow{EF}$$ and $$\overrightarrow{FG}$$ are parallel.