A vector carries a direction and a size. In space it takes three components, ⟨a, b, c⟩. Vectors add and scale component by component. The magnitude of ⟨a, b, c⟩ is √(a² + b² + c²), the Pythagorean theorem used twice. Dividing a vector by its magnitude leaves a unit vector pointing the same way.
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. The point (1, 2, 3) is 1 along, 2 across, 3 up.
Adding
⟨3, 4, 0⟩ + ⟨−1, 2, 5⟩ = ⟨2, 6, 5⟩. Geometrically the second vector starts where the first ends, and the sum runs from the first tail to the second tip. Nothing mixes between components.
Scaling
2⟨3, 4, 0⟩ = ⟨6, 8, 0⟩: twice as long, same direction. −⟨3, 4, 0⟩ = ⟨−3, −4, 0⟩: reversed. The direction line never changes; only the length and the sense do.
Why Pythagoras twice
The shadow of ⟨a, b, c⟩ on the floor has length √(a² + b²). The vector is the hypotenuse of a right triangle with that shadow and the height c: √(a² + b² + c²). |⟨3, 4, 12⟩| = √(9 + 16 + 144) = 13.
The vector between two points
From P(1, 2, 3) to Q(4, 6, 3): ⟨4 − 1, 6 − 2, 3 − 3⟩ = ⟨3, 4, 0⟩, end minus start, of length 5. The distance between the points is the magnitude of that vector.
Unit vectors
i = ⟨1, 0, 0⟩, j = ⟨0, 1, 0⟩, k = ⟨0, 0, 1⟩. Any vector is a combination: ⟨2, 3, 6⟩ = 2i + 3j + 6k. A unit vector isolates direction from size, which is what a direction of travel or a normal needs.
Step 2: Try It Yourself
Tap and try it out.
P(1, 2, 3), Q(4, 6, 3). Find the vector PQ, its length, a unit vector along it, and 2PQ − ⟨1, 1, 1⟩
- 1PQ = ⟨4 − 1, 6 − 2, 3 − 3⟩ = ⟨3, 4, 0⟩end minus start
Step 0 of 4
- 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: In Real Life
A drone’s position
A drone’s position is (x, y, z): east, north, up. Its velocity is a 3D vector. Adding, scaling and finding magnitude work exactly as in two dimensions, with one more component.
Step 4: 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, adds them to get 49, and takes the root: the magnitude is 7.
Step 5: Your Turn
Practice makes it stick.
The Drone
Problem 1 of 2
A drone moves ⟨3, 0, 4⟩ meters. How far did it travel from the start, in meters?
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?
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 6: 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; end minus start gives the vector between two points.
- The magnitude of ⟨a, b, c⟩ is √(a² + b² + c²), Pythagoras used twice.
- Divide by the magnitude whenever you only care about direction.