A vector carries a direction and a size. In space it takes three components instead of two: ⟨a, b, c⟩.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The third axis
The z-axis rises out of the xy-plane. Point the right hand along x and curl toward y, and the thumb gives z.
Adding
Vectors add component by component. Geometrically the second vector starts where the first ends.
Scaling
Multiplying by a number stretches the vector. A negative number also reverses it. The direction line never changes.
Magnitude
The length of ⟨a, b, c⟩ is the square root of a² + b² + c². It is the Pythagorean theorem used twice.
Unit vectors
Dividing a vector by its own magnitude leaves a vector of length 1 pointing the same way. That isolates direction from size.
A third coordinate changes little
A point in space is ⟨x, y, z⟩ and the magnitude is √(x² + y² + z²) — Pythagoras applied twice. Addition, scalar multiplication and the distance formula all extend componentwise without modification.
Visualising is the hard part
The algebra generalises effortlessly and the pictures do not. Sketching in three dimensions on paper is genuinely difficult, which is why level curves and cross sections become such important tools.
Orientation is a convention
The standard axes are right-handed: curling the right hand from x to y points the thumb along z. It is a choice, but a universal one, and cross products depend on it.
Unit vectors carry direction alone
Dividing by the magnitude gives a vector of length 1 pointing the same way. Separating direction from size is what makes directional derivatives and normal vectors work cleanly.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 4) · length 5
- Vector b(-1, 2) · length 2.24
- a + b(2, 6) · length 6.32
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 Amara Find a Unit Vector
Amara wants a unit vector in the direction of v = ⟨2, 3, 6⟩.
- Step 1
She squares the components: 4, 9 and 36.
Step 4: Your Turn
Practice makes it stick.
The Drone
Problem 1 of 2
A drone moves ⟨3, 0, 4⟩ metres. How far did it travel from the start, in metres?
The Two Pulls
Problem 2 of 2
One rope pulls ⟨4, 1, 0⟩ and another pulls ⟨−1, 2, 0⟩. What is the x-component of the total?
Work With Components
1 of 8
What is the magnitude of ⟨6, 8, 0⟩?
2 of 8
What is the magnitude of ⟨1, 2, 2⟩?
3 of 8
⟨2, 5, 1⟩ + ⟨3, −5, 4⟩. What is the y-component?
4 of 8
3⟨2, −1, 4⟩. What is the z-component?
5 of 8
A vector has magnitude 12. What is the magnitude of the unit vector in that direction?
6 of 8
⟨0, 0, 7⟩ points along which axis? Enter 1 for x, 2 for y, 3 for z.
7 of 8
Sort each expression by what it produces.
Tap something to move it.
- Empty
- Empty
8 of 8
The distance from (1, 2, 3) to (1, 6, 6). What is it?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is the magnitude of ⟨3, 4, 12⟩?
Question 2 of 2
What does multiplying a vector by −2 do?
What You Learned
- A vector in space has three components and carries direction plus size.
- Vectors add and scale component by component.
- The magnitude of ⟨a, b, c⟩ is the square root of a² + b² + c².