A vector carries both a size and a direction. Velocity and force are vectors; speed and mass are not.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Components
A vector is written by components, such as (3, 4), meaning 3 across and 4 up.
Magnitude
The magnitude is √(x² + y²), which is Pythagoras applied to the components.
Adding
Add vectors by components. Drawing them head to tail shows why: the resultant closes the triangle.
Magnitudes do not add
3 east and 4 north give a resultant of 5, not 7. Direction is part of the arithmetic.
Scalar multiplication
Multiplying by a number scales the length and leaves the direction alone, unless the number is negative, which reverses it.
Magnitude and direction
A vector records how far and in what direction, with no fixed starting point. Two arrows of equal length and direction anywhere on the page are the same vector.
Components, magnitude, direction
A vector ⟨a, b⟩ has magnitude √(a² + b²) and direction given by the inverse tangent, adjusted for quadrant. Moving between component form and magnitude-direction form is a routine requirement.
Adding and scaling
Vectors add component by component, which geometrically means nose to tail. Scalar multiplication stretches or, if negative, reverses. Both operations correspond directly to physical combinations of quantities.
Vectors and parametric motion
A parametric curve's position is a vector-valued function of t. Velocity is the vector of the coordinate rates. This is the link between the two topics and the entry point to vector calculus.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 0) · length 3
- Vector b(0, 4) · length 4
- a + b(3, 4) · length 5
The dashed arrow is b again, moved to the tip of a. The sum closes the triangle, and its components are just the x parts added and the y parts added.
Step 3: Watch an Example
One step at a time.
Watch Ines Add Two Vectors
Ines adds (3, 0) and (0, 4).
- Step 1
Adding components gives (3 + 0, 0 + 4) = (3, 4).
Step 4: Your Turn
Practice makes it stick.
The Magnitude
Problem 1 of 2
What is the magnitude of the vector (3, 4)?
The Sum
Problem 2 of 2
(2, 3) + (5, 1). What is the x component of the sum?
Work With Vectors
1 of 8
Magnitude of (6, 8)?
2 of 8
Magnitude of (5, 12)?
3 of 8
(1, 5) + (4, 2). What is the y component?
4 of 8
4 times the vector (3, 2). What is the y component?
5 of 8
Magnitude of (0, 9)?
6 of 8
Is mass a vector? 1 yes, 0 no.
7 of 8
Sort each quantity by whether it is a vector.
Tap something to move it.
- Empty
- Empty
8 of 8
Magnitude of (8, 6)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is the magnitude of (9, 12)?
Question 2 of 2
Why can magnitudes not simply be added?
What You Learned
- A vector has both magnitude and direction.
- Add vectors by components; magnitudes do not add.
- The magnitude is √(x² + y²), by Pythagoras.