Addition and Subtraction of Vectors

8.2

Addition and Subtraction of Vectors

Addition and Subtraction of Parallel Vector

Addition of Two Parallel Vector

The addition of two parallel vectors \(\utilde{a}\) and \(\utilde{b}\) can be combined to produce a resultant vector represented as \(\utilde{a}+\utilde{b}\).

For example:

Kalkulator yang menunjukkan dua persamaan yang berbeza, mencontohkan proses penambahan vektor a dan vektor b dalam matematik.

Subtraction of Two Parallel Vector

The subtraction of vector \(\utilde{b}\) from vector \(\utilde{a}\) is the sum of vector \(\utilde{b}\) and negative vector \(\utilde{b}\), that is \(\utilde{a}-\utilde{b}=\utilde{a}+(-\utilde{b})\).

For example:

Graf yang menggambarkan pelbagai jenis persamaan, memaparkan contoh penolakan antara vektor a dan vektor b.

Addition and Subtraction of Non-Parallel Vector

Triangle Law

A triangle illustrating the addition of non-parallel vectors a and b, featuring two sides and a central point.

Parallelogram Law

A triangle with two sides and a point at the bottom, illustrating the addition of non-parallel vectors in parallelogram form.

Polygon Law

Diagram of a trapezium with labeled sides, illustrating the addition of non-parallel vectors a, b, and c in polygon form.

Example \(1\)

Question

It is given that,

\(\overrightarrow{CD}=3\utilde{r}\),
\(\overrightarrow{EF}=5\utilde{r}\),
\(\overrightarrow{FG}=\utilde{r}\).

If \(|\utilde{r}|=3\) units, find the magnitude of the following expression:

\(\begin{aligned} 2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \end{aligned}\).

Solution

Express the expression in term of \(\utilde{r}\):

\(\begin{aligned} &2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \\\\ &=2(3\utilde{r})+5 \utilde{r}-3(\utilde{r}) \\\\ &=6\utilde{r}+5 \utilde{r}-3\utilde{r}. \end{aligned}\)

Since \(|\utilde{r}|=3\) units, hence:

\(\begin{aligned} &|6\utilde{r}+5 \utilde{r}-3\utilde{r}|\\\\ &=|8\utilde{r}|\\\\&=8|\utilde{r}|\\\\ &=8(3) \\\\ &=24 \text{ units}. \end{aligned}\)

Example \(2\)

Question

Two forces \(\bold{F}_1=50 \text{ N}\) and \(\bold{F}_2=30 \text{ N}\) are acting on a moving object.

Determine the magnitude of the force and the direction of the movement of the object if,

\(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction.

Solution

If \(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction, then:

A square with two red circles, illustrating opposing forces F1 and F2 acting on the square object.

Magnitude:

\(\begin{aligned} \bold{F}_1+(-\bold{F}_2)&=50+(-30)\\\\&=50-30 \\\\ &=20\text{ N}. \end{aligned}\)

The magnitude of the force acted on the object is \(20\text{ N}\) in the same dircetion as \(\bold{F}_1\).