Vectors &
Velocity
Master how to represent speed and direction as vectors, combine them with addition, and solve real-world velocity problems using components and magnitude.
What Is a Vector?
A vector is a quantity that has both magnitude (size) and direction. Unlike a scalar (which has only magnitude), a vector tells you how much and which way.
Visualizing Vectors as Arrows
An arrow represents a vector: the tail is the starting point, the tip is the ending point. The arrow's length encodes magnitude and its direction encodes the angle.
Why Velocity Is a Vector
Speed tells you how fast something moves. Velocity tells you how fast and in which direction. That extra piece of information—direction—is what makes velocity a vector quantity.
Component Form of a Vector
Every 2D vector can be broken into horizontal and vertical components. If the vector has magnitude \(r\) and direction angle \(\theta\) (measured from the positive \(x\)-axis):
Magnitude and Direction
Given a vector in component form \(\vec{v} = \langle a,\, b \rangle\), we recover its magnitude and direction angle:
Adding Vectors
To add two vectors, add their corresponding components:
Graphically, place the tail of \(\vec{v}\) at the tip of \(\vec{u}\). The resultant goes from the tail of \(\vec{u}\) to the tip of \(\vec{v}\): the tip-to-tail method.
Scalar × Vector vs. Vector + Vector
| Scalar Multiplication | Vector Addition | |
|---|---|---|
| Operation | \(k\vec{v} = \langle ka,\, kb \rangle\) | \(\vec{u}+\vec{v} = \langle u_x+v_x,\, u_y+v_y \rangle\) |
| Effect | Scales magnitude; flips direction if \(k < 0\) | Combines two vectors into one resultant |
| Result type | Vector (same or opposite direction) | Vector (new direction & magnitude) |
| Example | \(3\langle 2,1 \rangle = \langle 6,3 \rangle\) | \(\langle 2,1 \rangle + \langle 1,3 \rangle = \langle 3,4 \rangle\) |
How to Solve a Velocity Problem
Example 1: Boat Crossing a River
A boat travels at 8 mph due east. A river current pushes at 6 mph due south. Find the resultant velocity.
Example 2: Wind and an Airplane
A plane's airspeed is 250 mph heading \(60°\) above the positive \(x\)-axis. Wind blows at 40 mph at \(180°\) (due west). Find the ground velocity.
Example 3: Two Tugboats
Two tugboats pull a barge. Tug A pulls with force 5000 N at \(30°\). Tug B pulls with force 7000 N at \(120°\). Find the resultant.
Unit Vectors \(\hat{i}\) and \(\hat{j}\)
The standard unit vectors are \(\hat{i} = \langle 1,0 \rangle\) and \(\hat{j} = \langle 0,1 \rangle\). Any vector \(\langle a,b \rangle\) can be written:
A unit vector in the direction of \(\vec{v}\) is:
Bearing Angles vs. Standard Angles
Common Mistakes
Key Formulas at a Glance
| Concept | Formula |
|---|---|
| Component form | \(\langle r\cos\theta,\, r\sin\theta \rangle\) |
| Magnitude | \(\|\vec{v}\| = \sqrt{a^2 + b^2}\) |
| Direction | \(\theta = \tan^{-1}\!\left(\frac{b}{a}\right)\) |
| Vector addition | \(\langle u_x+v_x,\, u_y+v_y \rangle\) |
| Scalar multiplication | \(k\langle a,b \rangle = \langle ka,kb \rangle\) |
| Unit vector | \(\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}\) |
| Bearing to standard | \(\theta = 90° - \beta\) |
Question 1 of 5
A car drives at 40 mph due east and encounters a crosswind of 30 mph due north. What is the car's resultant speed?
Question 2 of 5
A velocity vector has components \(\langle -5, 12 \rangle\). What is its magnitude?
Question 3 of 5
Which expression gives the \(x\)-component of a velocity vector with magnitude \(r\) and direction angle \(\theta\)?
Question 4 of 5
A plane flies on a bearing of \(N\,40°\,E\). What is the standard angle \(\theta\) (measured counter-clockwise from the positive \(x\)-axis)?
Question 5 of 5
If \(\vec{u} = \langle 3, 4 \rangle\) and \(\vec{v} = \langle -1, 2 \rangle\), what is \(\vec{u} + \vec{v}\)?