A vector carries size and direction. A speed of 30 is a scalar; 30 heading north is a vector.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Components
Writing a vector as an x part and a y part turns every operation into ordinary arithmetic.
Adding
Add the x parts and add the y parts, separately. Nothing mixes between them.
Magnitude
The length is the hypotenuse of the two components, by Pythagoras.
Magnitude and direction, no position
A vector records how far and in what direction, but not from where. Two arrows of the same length and direction anywhere on the page represent the same vector, which is what distinguishes vectors from points.
Components and magnitude
A vector in the plane is written ⟨a, b⟩, and its magnitude is √(a² + b²) by Pythagoras. Direction is usually given as an angle from the positive x-axis, found with the inverse tangent and adjusted for quadrant.
Addition and scalar multiplication
Vectors add component by component, which geometrically means nose to tail. Multiplying by a scalar stretches or shrinks and, if negative, reverses. Both operations have immediate physical meanings.
Unit vectors carry direction alone
Dividing a vector by its magnitude gives a vector of length 1 pointing the same way. That separates direction from size, which is exactly what you want when specifying a heading or a normal direction.
Step 2: Try It Yourself
Tap and try it out.
The hypotenuse is not given. It comes from 3² + 4² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Yusuf Add Two Vectors
Yusuf adds (3, 4) and (1, 2), then finds the magnitude of the first.
- Step 1
The x parts add: 3 + 1 = 4.
Step 4: Your Turn
Practice makes it stick.
The Sum
Problem 1 of 2
(2, 5) + (6, 1). What is the x component of the sum?
The Length
Problem 2 of 2
What is the magnitude of (6, 8)?
Components and Length
1 of 8
(1, 2) + (3, 4). x component of the sum?
2 of 8
Same sum. y component?
3 of 8
Magnitude of (3, 4)?
4 of 8
Magnitude of (5, 12)?
5 of 8
Scalar multiple: 3 times (2, 5). x component?
6 of 8
Is speed a vector? 1 for yes, 0 for no.
7 of 8
Magnitude of (0, 7)?
8 of 8
Select every quantity that is a vector.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Magnitude of (9, 12)?
What You Learned
- A vector has magnitude and direction; a scalar has only magnitude.
- Vectors add component by component, with nothing mixing between them.
- The magnitude is the hypotenuse of the components.