Skill: Resolve vectors into components

A vector can be split into two perpendicular parts:

• a horizontal component, $V_x$
• a vertical component, $V_y$

These components add together to reproduce the original vector. Their signs show direction:

• $+$ horizontal: right
• $-$ horizontal: left
• $+$ vertical: up
• $-$ vertical: down

Suppose a vector has magnitude $V$ and makes an angle $\theta$ measured counterclockwise from the positive horizontal axis. Then

$$V_x=V\cos\theta \qquad\text{and}\qquad V_y=V\sin\theta.$$

The signs come from the vector’s direction.

Example

Resolve a $20\text{ N}$ force directed $40^\circ$ north of west into horizontal and vertical components.

Step 1: Identify the direction

“$40^\circ$ north of west” means the vector points:

• left, so its horizontal component is negative;
• up, so its vertical component is positive.

The corresponding standard angle from the positive horizontal axis is

$$\theta=180^\circ-40^\circ=140^\circ.$$

Step 2: Find the horizontal component

$$F_x=F\cos\theta$$ $$F_x=20\cos(140^\circ)\approx -15.3\text{ N}.$$

The negative sign means $15.3\text{ N}$ to the left. It does not mean a negative length.

Step 3: Find the vertical component

$$F_y=F\sin\theta$$ $$F_y=20\sin(140^\circ)\approx 12.9\text{ N}.$$

The positive sign means $12.9\text{ N}$ upward.

Step 4: State the components

The force has components

$$\boxed{F_x\approx -15.3\text{ N}} \qquad\text{and}\qquad \boxed{F_y\approx 12.9\text{ N}}.$$

So the original force can be written as

$$\vec F\approx \langle -15.3,\ 12.9\rangle\text{ N}.$$

The two components are perpendicular, and together they reproduce the original $20\text{ N}$ force.