Vector Applications

Vectors are incredibly useful for modeling real-world situations where quantities have both a magnitude (size) and a direction. The most common applications of vectors in physics and mathematics involve forces, velocities, and navigation.

Resolving Vectors into Components

Often, you will know the magnitude and direction of a single vector, but you need to know its horizontal and vertical effects. This is called resolving a vector into its components.

If a vector $\vec{v}$ has a magnitude $||\vec{v}||$ and makes an angle $\theta$ with the horizontal, its components are: $v_x = ||\vec{v}|| \cos(\theta)$ $v_y = ||\vec{v}|| \sin(\theta)$

Example: A 100-lb force acts at an angle of $30^\circ$ from the horizontal. Find the horizontal and vertical components.

• Horizontal force: $F_x = 100 \cos(30^\circ) = 100 \left(\frac{\sqrt{3}}{2}\right) = 50\sqrt{3}$ lbs.
• Vertical force: $F_y = 100 \sin(30^\circ) = 100 \left(\frac{1}{2}\right) = 50$ lbs.

Resultant Vectors

When multiple forces or motions act on an object simultaneously, their combined effect is called the resultant vector. You can find the resultant by adding the individual vectors together.

Example: A boat heads east at 20 mph while the current flows north at 5 mph. Find the resultant speed and direction.

1. Write the vectors: The boat's velocity is $\vec{B} = \langle 20, 0 \rangle$ and the current's velocity is $\vec{C} = \langle 0, 5 \rangle$.
2. Add them: The resultant velocity $\vec{R} = \vec{B} + \vec{C} = \langle 20, 5 \rangle$.
3. Find the speed (magnitude): $||\vec{R}|| = \sqrt{20^2 + 5^2} = \sqrt{400 + 25} = \sqrt{425} \approx 20.6 \text{ mph}$
4. Find the direction: Use the tangent function to find the angle. $\tan(\theta) = \frac{5}{20} = 0.25$ $\theta = \tan^{-1}(0.25) \approx 14.04^\circ$ The boat is traveling at approximately 20.6 mph at an angle of $14.04^\circ$ north of east.

Equilibrium

An object is in a state of equilibrium when the sum of all forces (the resultant vector) acting on it is zero. If you have several forces acting on an object and it isn't moving, you know that their vector sum must equal $\langle 0, 0 \rangle$. This principle is widely used to calculate unknown tensions in cables or structural supports.