Vector

\(\blacksquare\) 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.

\(\blacksquare\) 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 \(\utilde{a}\).

Example 1:

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:

Given a pair of vector.

Determine whether the pair of vector is parallel or not.

\(\overrightarrow{EF}=\dfrac{1}{3}\utilde{r}\)

\(\overrightarrow{FG}=9\utilde{r}\)

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,

\(\utilde{r}=3\overrightarrow{EF}\).

\(\therefore \overrightarrow{EF}\) and \(\overrightarrow{FG}\) are parallel.