Vector Addition and Subtraction with Analytical Method

Calculating a Resultant with Components

The graphical method helps us see the resultant direction, but its accuracy depends on the drawing. The analytical method is more precise because it uses component calculations.

The method is to resolve every vector into the $x$ and $y$ axes, add the components along each axis, and then rebuild the resultant.

R_x=\sum F_x

In the analytical method, vectors are added axis by axis, not directly from their arrow lengths.

Steps for Vector Addition

Suppose several forces $\vec{F}_1$ , $\vec{F}_2$ , and $\vec{F}_3$ act on an object. Each force is first split into horizontal and vertical components.

After all components are known, the resultant components are:

R_x=F_{1x}+F_{2x}+F_{3x}+\cdots

The resultant magnitude is:

R=\sqrt{R_x^2+R_y^2}

The direction of the resultant relative to the positive $x$ axis can be found from:

\tan\theta=\frac{R_y}{R_x}

If $R_x$ or $R_y$ is negative, identify the quadrant first so the direction is not misplaced.

Two-Stage Displacement

A person walks in two stages. The first stage is equivalent to $80\text{ m}$ east and $60\text{ m}$ north. The second stage is equivalent to $120\text{ m}$ east and $90\text{ m}$ north.

Displacement vector	$x$ component	$y$ component
$\vec{s}_1$	$80\text{ m}$	$60\text{ m}$
$\vec{s}_2$	$120\text{ m}$	$90\text{ m}$
$\vec{R}$	$200\text{ m}$	$150\text{ m}$

The resultant components are:

R_x=80+120=200\text{ m}

The total displacement magnitude is:

\begin{aligned} R&=\sqrt{200^2+150^2} \\ &=\sqrt{40000+22500} \\ &=250\text{ m} \end{aligned}

The resultant direction is:

\tan\theta=\frac{150}{200}=0.75

Therefore $\theta\approx36.9^\circ$ above east. The total displacement is $250\text{ m}$ at $36.9^\circ$ north of east.

Subtraction with Components

Vector subtraction is also done component by component. If $\vec{C}=\vec{A}-\vec{B}$ , then:

C_x=A_x-B_x

For example, let $\vec{A}=7\hat{i}+4\hat{j}$ and $\vec{B}=2\hat{i}+6\hat{j}$ .

\begin{aligned} \vec{C}&=(7-2)\hat{i}+(4-6)\hat{j} \\ &=5\hat{i}-2\hat{j} \end{aligned}

The negative sign in $-2\hat{j}$ means the resulting $y$ component points downward.

So subtraction does not need a new rule. Keep the same axes, subtract matching components, then read the signs to recover the direction of the result.