Applications and Vectors

You have spent time learning about sine, cosine, and tangent. You can find missing sides and angles in triangles. You understand the unit circle. But here is the question that might be nagging at you: when am I ever going to use this?

The answer is: all the time, in ways you might not expect. Every time a pilot navigates through the sky, every time an engineer calculates whether a bridge can handle the load, every time a physicist predicts where a ball will land, trigonometry is doing the heavy lifting. And at the heart of many of these applications is a concept called a vector.

So, if the word “vector” seems scary to you, then know that you are not alone. Many students hear the word and imagine something complicated and abstract. But you need to realize something: you already know how to work with vectors. You may not believe it, but, really, you do. If you have ever given directions like “go 3 blocks north, then 2 blocks east,” you have described vectors. If you have ever felt the wind push you sideways while you are walking forward, you have experienced vector addition. This lesson is about taking that intuition and making it precise.

Core Concepts

Bearings and Navigation

Before GPS, sailors and pilots needed a precise way to describe direction. They developed the system of bearings, which is still used today in aviation, marine navigation, and surveying.

A bearing tells you which direction to travel, measured as an angle from a reference direction. There are two common ways to express bearings:

Compass Bearings: These use cardinal directions (N, S, E, W) as references. You start at either North or South, then specify the angle toward East or West.

• N 30° E means: start facing North, rotate 30° toward the East
• S 45° W means: start facing South, rotate 45° toward the West
• N 0° E is the same as due North
• N 90° E is the same as due East

True Bearings (Three-Figure Bearings): These measure the angle clockwise from North, always using three digits.

• 000° is due North
• 090° is due East
• 180° is due South
• 270° is due West
• 045° is Northeast (N 45° E in compass bearing)

True bearings are common in aviation because they are unambiguous: every direction corresponds to exactly one number between 000° and 360°.

The key to solving navigation problems is drawing a clear diagram. Always:

1. Draw a vertical line representing North-South
2. Mark the bearing angle from North
3. Use right triangles to find north-south and east-west components

Angle of Elevation and Depression

When you look up at something above you, the angle your line of sight makes with the horizontal is called the angle of elevation. When you look down at something below you, that angle is called the angle of depression.

Here is the crucial insight: the angle of elevation from point A to point B equals the angle of depression from point B to point A. They are alternate interior angles formed by a horizontal line and the line of sight, with a transversal (your line of sight) crossing them.

These angles appear constantly in real-world problems:

• Surveyors measuring the height of a building
• Pilots calculating descent angles
• Engineers designing wheelchair ramps
• Astronomers measuring the position of stars above the horizon

The setup is always the same: you have a right triangle where one angle is the angle of elevation or depression, one side is a known distance (often horizontal or vertical), and you need to find another side using trigonometric ratios.

Introduction to Vectors

A scalar is a quantity that has only magnitude (size). Temperature, mass, and time are scalars. When you say “it took 3 hours,” you have completely described the time.

A vector is a quantity that has both magnitude and direction. When you say “the wind is blowing at 20 mph,” you have not given complete information. From which direction? North? Southwest? The direction matters. Velocity, force, and displacement are all vectors.

Vectors are typically represented in two ways:

Geometric representation: An arrow where the length represents magnitude and the direction of the arrow represents, well, direction. We often name vectors with bold letters like v or with an arrow above them like $\vec{v}$.

Component form: A vector can be written as $\mathbf{v} = \langle a, b \rangle$, where $a$ is the horizontal component and $b$ is the vertical component. Think of this as: “go $a$ units in the x-direction and $b$ units in the y-direction.”

The vector $\langle 3, 4 \rangle$ means: move 3 units right and 4 units up.

Magnitude and Direction

The magnitude of a vector $\mathbf{v} = \langle a, b \rangle$ is its length, written as $|\mathbf{v}|$ or $|\mathbf{v}|$. By the Pythagorean Theorem:

$$|\mathbf{v}| = \sqrt{a^2 + b^2}$$

For example, the vector $\langle 3, 4 \rangle$ has magnitude $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

The direction of a vector is the angle it makes with the positive x-axis, measured counterclockwise. If $\theta$ is this angle, then:

$$\tan \theta = \frac{b}{a}$$

So $\theta = \tan^{-1}\left(\frac{b}{a}\right)$, but you need to be careful about which quadrant the vector is in. The arctangent function only gives angles in the first and fourth quadrants, so you may need to add 180° if the vector points into the second or third quadrant.

Converting Between Forms

Often you will need to convert between the two representations.

From magnitude and direction to component form:

If a vector has magnitude $r$ and makes angle $\theta$ with the positive x-axis:

$$a = r \cos \theta$$ $$b = r \sin \theta$$

So $\mathbf{v} = \langle r \cos \theta, r \sin \theta \rangle$.

This should feel familiar: it is exactly how you found the coordinates of a point on the unit circle, just scaled by $r$.

From component form to magnitude and direction:

Given $\mathbf{v} = \langle a, b \rangle$:

$$r = |\mathbf{v}| = \sqrt{a^2 + b^2}$$ $$\theta = \tan^{-1}\left(\frac{b}{a}\right) \text{ (adjusted for quadrant)}$$

Adding Vectors

This is where vectors become powerful. To add two vectors, you simply add their corresponding components:

$$\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$$

Geometrically, this is like placing the tail of the second vector at the head of the first. The sum (called the resultant) goes from the tail of the first to the head of the second. This is sometimes called the “tip-to-tail” method.

Alternatively, if both vectors start at the same point, the resultant is the diagonal of the parallelogram they form. This is the “parallelogram method.”

Both methods give the same answer, just visualized differently.

Subtracting vectors works similarly:

$$\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$$

Scalar multiplication scales a vector:

$$k \langle a, b \rangle = \langle ka, kb \rangle$$

If $k > 0$, the vector points in the same direction but is longer or shorter. If $k < 0$, the vector reverses direction.

Applications: Force, Velocity, and Displacement

Displacement is the change in position from one point to another. If you walk 5 meters east, your displacement is the vector $\langle 5, 0 \rangle$ (assuming east is positive x). If you then walk 3 meters north, that is $\langle 0, 3 \rangle$. Your total displacement is $\langle 5, 0 \rangle + \langle 0, 3 \rangle = \langle 5, 3 \rangle$.

Velocity is speed with a direction. A car going 60 mph north has a different velocity than a car going 60 mph east, even though they have the same speed. Velocities add as vectors: if you are on a boat going 10 mph east in a river flowing 3 mph south, your actual velocity relative to the ground is $\langle 10, -3 \rangle$ (if east is positive x and north is positive y).

Force is a push or pull with both magnitude and direction. When multiple forces act on an object, you add them as vectors to find the net force or resultant force. This is fundamental to all of physics and engineering.

Simple Harmonic Motion

Many things in nature oscillate: a pendulum swings back and forth, a spring bounces up and down, a guitar string vibrates, the tides rise and fall. When the motion follows a sine or cosine pattern, we call it simple harmonic motion (SHM).

The position of an object in SHM can be described by:

$$y = A \cos(\omega t)$$ or $$y = A \sin(\omega t)$$

where:

• $A$ is the amplitude: the maximum displacement from equilibrium (how far it swings)
• $\omega$ (omega) is the angular frequency: how fast it oscillates, in radians per unit time
• $t$ is time

The period $T$ is the time for one complete cycle:

$$T = \frac{2\pi}{\omega}$$

The frequency $f$ is how many cycles per unit time:

$$f = \frac{1}{T} = \frac{\omega}{2\pi}$$

Which function you use (sine or cosine) depends on the initial conditions:

• Use $\cos$ if the object starts at maximum displacement (like a pendulum released from the side)
• Use $\sin$ if the object starts at the equilibrium position moving in the positive direction

You can also add a phase shift $\phi$ to account for different starting positions:

$$y = A \cos(\omega t + \phi)$$

This is why trigonometry is so important in physics: the same mathematical functions that describe triangles also describe oscillations, waves, and countless other phenomena.

Notation and Terminology

Term	Meaning	Example
Bearing	Direction measured from north	N 30° E, or 030°
Angle of elevation	Angle upward from horizontal	Looking up at a plane
Angle of depression	Angle downward from horizontal	Looking down from a cliff
Vector	Quantity with magnitude and direction	Force, velocity
Scalar	Quantity with magnitude only	Speed, temperature
Component form	$\mathbf{v} = \langle a, b \rangle$	$\langle 3, 4 \rangle$
Magnitude	Length of a vector: $	\mathbf{v}
Resultant	Sum of two or more vectors	Net force
Amplitude	Maximum displacement in SHM	$A$ in $y = A\cos(\omega t)$
Period	Time for one complete oscillation	$T = \frac{2\pi}{\omega}$
Angular frequency	Rate of oscillation in radians per time	$\omega$ in $y = A\cos(\omega t)$

Examples

Key Properties and Rules

Vector Operations

Addition: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$

Subtraction: $\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$

Scalar multiplication: $k\langle a, b \rangle = \langle ka, kb \rangle$

Magnitude: $|\langle a, b \rangle| = \sqrt{a^2 + b^2}$

Converting Between Forms

Have	Want	Use
Magnitude $r$, angle $\theta$	Components	$a = r\cos\theta$, $b = r\sin\theta$
Components $a$, $b$	Magnitude	$r = \sqrt{a^2 + b^2}$
Components $a$, $b$	Direction	$\theta = \tan^{-1}(b/a)$ (adjust for quadrant)

Bearing Conversions

For a bearing of $\beta$ (measured clockwise from north):

• East component: distance $\times \sin \beta$
• North component: distance $\times \cos \beta$

Simple Harmonic Motion

$$y = A\cos(\omega t) \quad \text{or} \quad y = A\sin(\omega t)$$

Quantity	Symbol	Formula
Amplitude	$A$	Maximum displacement
Angular frequency	$\omega$	$\omega = \frac{2\pi}{T} = 2\pi f$
Period	$T$	$T = \frac{2\pi}{\omega}$
Frequency	$f$	$f = \frac{1}{T} = \frac{\omega}{2\pi}$

Initial conditions:

• Object starts at maximum displacement: use $y = A\cos(\omega t)$
• Object starts at equilibrium, moving positive: use $y = A\sin(\omega t)$

Common Mistakes to Avoid

• Confusing bearing angles with standard position angles: Bearings are measured clockwise from north; standard position angles are counterclockwise from east.
• Forgetting that “wind from” means the opposite direction: Wind “from the north” blows toward the south.
• Not adjusting arctangent for quadrant: $\tan^{-1}$ only gives angles in Q1 and Q4. If the vector is in Q2 or Q3, add 180°.
• Confusing speed and velocity: Speed is a scalar (just magnitude); velocity is a vector (magnitude and direction).
• Mixing up period and frequency: Period is time per cycle (seconds); frequency is cycles per time (Hz). They are reciprocals.

Real-World Applications

Aviation Navigation

Pilots constantly deal with the difference between heading (where the nose points) and track (where the plane actually goes). Crosswinds push planes off course, and pilots must calculate the “wind correction angle” to maintain their intended track. This is exactly the vector addition problem from Example 6.

Flight computers and autopilots do these trigonometric calculations automatically, but pilots are trained to do them by hand as a backup. The E6B “whiz wheel,” a circular slide rule that has been standard pilot equipment for decades, is essentially a mechanical vector calculator.

Ocean Currents and Sailing

Sailors face the same challenge as pilots, but with ocean currents instead of wind. A boat that can travel at 8 knots through the water might only make 6 knots over the ground if it is fighting a 2-knot current, or it might make 10 knots with a following current.

Modern navigation systems show both speed through water (from the boat’s instruments) and speed over ground (from GPS), and the difference tells you about the current. Competitive sailors and racing teams use detailed current charts and vector calculations to find the fastest route.

Physics: Projectile Motion and Forces

When you throw a ball at an angle, its initial velocity is a vector. Physics uses trigonometry to break this into horizontal and vertical components, which then evolve independently (the horizontal component stays constant while the vertical component changes due to gravity).

Similarly, forces in physics always combine as vectors. An object on a ramp experiences gravitational force pulling straight down, but we break this into components parallel and perpendicular to the ramp to analyze the motion. Bridges, buildings, and machines are all designed using vector analysis of forces.

Engineering: Structural Analysis

Civil engineers must ensure that structures can handle all the forces acting on them. Wind loads, weight loads, and tension in cables all act as vectors. The joints in a bridge truss experience forces from multiple directions, and engineers use vector addition to find the total force each joint must withstand.

Even something as simple as a hanging sign involves vectors: the tension in each supporting cable must add up to a vector that exactly balances the weight of the sign.

Pendulums and Springs

The grandfather clock in a museum, the shock absorbers in your car, the vibrating atoms in a solid, and the electrical oscillations in a radio circuit all exhibit simple harmonic motion. The same equations describe them all.

In music, a vibrating guitar string undergoes SHM (approximately), and the frequency of vibration determines the pitch you hear. The amplitude determines the volume. Every time you listen to music, you are experiencing the trigonometry of simple harmonic motion translated into sound waves.

Self-Test Problems

Summary

• Bearings describe direction using angles from north. Compass bearings use N/S and E/W (like N 30° E), while true bearings use three-digit clockwise angles from north (like 030°). For a bearing $\beta$, the north component is distance $\times \cos\beta$ and the east component is distance $\times \sin\beta$.

• Angles of elevation and depression are measured from the horizontal. They are equal when measured from opposite ends of the same line of sight, and they set up right triangle problems that can be solved with basic trigonometry.

• Vectors have both magnitude and direction. They can be represented geometrically (as arrows) or in component form: $\mathbf{v} = \langle a, b \rangle$.

• The magnitude of $\langle a, b \rangle$ is $\sqrt{a^2 + b^2}$. The direction is found using $\tan^{-1}(b/a)$, adjusted for the correct quadrant.

• To convert from magnitude $r$ and angle $\theta$ to components: $a = r\cos\theta$ and $b = r\sin\theta$.

• Vectors add by adding their components: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$. This models how velocities combine, how forces add, and how displacements accumulate.

• Simple harmonic motion is described by $y = A\cos(\omega t)$ or $y = A\sin(\omega t)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and the period is $T = 2\pi/\omega$.

• These concepts have countless real-world applications: navigation (bearing problems and wind correction), physics (projectile motion, forces, oscillations), engineering (structural analysis), and more.

• The key to solving these problems is always the same: draw a diagram, identify the right triangle or vector relationship, and apply trigonometry. Whether you are calculating a pilot’s ground track or predicting where a pendulum will be in half a second, you are using the same fundamental tools.

You have now seen how the trigonometry you learned about angles and triangles extends to describe motion, forces, and navigation in the real world. Vectors are the language that physics and engineering use to describe quantities with direction, and simple harmonic motion is the pattern that describes everything from heartbeats to radio waves. These applications are why trigonometry is not just an abstract mathematical exercise but an essential tool for understanding how the world works.