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: Picture It
Start with something you already know.
The longest pencil in a small box
- A short pencil lies flat in a small box, corner to corner across the bottom.
- That short pencil and the two bottom edges make a right triangle.
- A longer pencil runs from the same corner up to the top corner right above where the short one ended.
- The flat diagonal and the box’s height make a second right triangle.
- That longer pencil is the long side of the second triangle, and it is the longest pencil the box can hold.
A vector in space is the longer pencil: its three components are the box’s length, width and height.
Its magnitude is Pythagoras twice: first across the floor, then up to the far corner.
A box’s edges are never negative, but a component can be. Its square is not, so the length works the same.
A tall, narrow box, 2 inches by 3 inches on the bottom and 6 inches tall:
| Measure | Value |
|---|---|
| Short pencil, squared | 2² + 3² = 13 |
| Longer pencil, squared | 13 + 6² = 49 |
| Longer pencil | √49 = 7 inches |
That is |⟨2, 3, 6⟩| = √(4 + 9 + 36) = 7: the floor first, then the height.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A drone lifts off a pad and flies 6 meters east and 8 meters north, staying just above the ground. How far is it from the pad in a straight line, in meters?
meters
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Sam Measure a Drone’s Trip
A drone flies in a straight line from P(3, −2, 5) to Q(7, 6, 6), in meters. Sam finds the trip as a vector, its length, and a unit vector pointing the same way.
- Step 1
The trip is end minus start, one component at a time: ⟨7 − 3, 6 − (−2), 6 − 5⟩ = ⟨4, 8, 1⟩. Subtracting −2 is adding 2, which is why the middle component is 8 and not 4.
Watch Ravi Combine Two Vectors, Then Finish It Yourself
u = ⟨1, 3, −2⟩ and v = ⟨0, 2, −8⟩. Ravi wants 2u − v, its length, and a unit vector along it. He starts; the rest is yours.
- Step 1
Scaling comes first. Multiplying u by 2 doubles every component, so 2u = ⟨2, 6, −4⟩. The direction stays the same and the length doubles.
Watch Noor Test Whether Two Vectors Point the Same Way
a = ⟨2, −4, 6⟩. Noor asks whether b = ⟨−3, 6, −9⟩ and c = ⟨3, −6, 8⟩ lie along the same line as a.
- Step 1
Scaling never changes the direction line, so two vectors share a line exactly when one is a number times the other. Noor divides matching components of b by a: −3/2 = −1.5, 6/(−4) = −1.5 and −9/6 = −1.5.
Step 6: Your Turn
Practice makes it stick.
Two Legs of a Flight
Problem 1 of 17
A drone flies ⟨5, −2, 4⟩ meters, then ⟨1, 5, −2⟩ meters (east, north, up). How far is it from its takeoff point in a straight line, in meters?
meters
The Zipline Cable
Problem 2 of 17
A zipline starts on a platform 10 meters up a tower and ends on the ground 40 meters east and 80 meters north of the tower’s base. With the base at (0, 0, 0), x east, y north and z up, the cable runs from (0, 0, 10) to (40, 80, 0). Cable costs $15 a meter. What does the straight cable cost, in dollars?
dollars
Work With Components
One component at a time, then Pythagoras twice where a length is asked for.
Problem 3 of 17
What is the magnitude of ⟨2, 10, 11⟩?
Problem 4 of 17
What is the distance between A(1, −2, 3) and B(7, 4, 10)?
Problem 5 of 17
u = ⟨3, −1, 2⟩ and v = ⟨−5, 4, 6⟩. What is the y-component of 3u + v?
Problem 6 of 17
What is the y-component of the unit vector in the direction of ⟨0, −8, 6⟩? Give it as a decimal.
Problem 7 of 17
What is the magnitude of 8i − j + 4k?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 17
Priya found the magnitude of ⟨−4, 7, 4⟩. One line is a slip. Which?
Priya’s magnitude of ⟨−4, 7, 4⟩
- 1|v| = √((−4)² + 7² + 4²)
- 2= √(−16 + 49 + 16)
- 3= √49
- Answer= 7
Problem 9 of 17
Worked correctly, what is the magnitude of ⟨−4, 7, 4⟩?
Problem 10 of 17
Dev found the vector from P(4, 1, −2) to Q(1, 5, 2). One line is a slip. Which?
Dev’s vector from P(4, 1, −2) to Q(1, 5, 2)
- 1PQ = Q − P
- 2PQ = ⟨4 − 1, 1 − 5, −2 − 2⟩
- AnswerPQ = ⟨3, −4, −4⟩
Problem 11 of 17
Worked correctly, what is the x-component of the vector from P(4, 1, −2) to Q(1, 5, 2)?
Problem 12 of 17
Maya wanted the unit vector in the direction of ⟨4, −4, 2⟩. One line is a slip. Which?
Maya’s unit vector along ⟨4, −4, 2⟩
- 1size: 4 + (−4) + 2 = 2
- 2unit = ⟨4, −4, 2⟩ ÷ 2
- Answerunit = ⟨2, −2, 1⟩
Problem 13 of 17
Worked correctly, what is the z-component of the unit vector in the direction of ⟨4, −4, 2⟩? Round to two decimal places.
Pick the Move
Choose the move that gets the answer. You do not need to work either question all the way through.
Problem 14 of 17
You want the distance between (2, 7, −1) and (3, 19, 11). Which move gets it?
Problem 15 of 17
A drone should fly exactly 10 meters in the direction of ⟨1, 2, 2⟩. Which move gives its trip as a vector?
Same Length or Same Direction?
Judge each one without a calculator. Select every entry that fits.
Problem 16 of 17
Select every vector that points in the same direction as ⟨2, −1, 3⟩.
Problem 17 of 17
Select every vector that has the same length as ⟨2, 3, 6⟩.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which positive number c does ⟨c, 6, 9⟩ have a length of exactly 11?
Stretch 2 of 2
P is (2, −1, 5) and Q is (8, 3, −1). The point M halfway from P to Q is P plus half of the vector PQ. What is M’s z-coordinate?
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.
For the grown-up
The misconception this lesson targets is that the third coordinate is something new, when it changes nothing about the rules: a student who holds it treats a 3D vector as a different kind of object and reaches for new formulas, then makes the old slips in a new place. The two that matter most are finding a length by adding the components instead of squaring them (or squaring a negative and keeping the minus sign), and building the vector between two points as start minus end. Both survive because the working looks tidy afterwards. The habits that settle them are cheap: say "end minus start" and check that start plus the vector lands on the end, and check any unit vector by confirming its squares add to 1. The box picture, across the floor and then up, is what makes Pythagoras twice feel inevitable rather than memorized.