The dot product multiplies matching components and adds: ⟨a₁, a₂, a₃⟩ · ⟨b₁, b₂, b₃⟩ = a₁b₁ + a₂b₂ + a₃b₃. The result is a number, not a vector.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What the number means
It measures how much the two vectors agree. Pointing the same way gives a large positive value; pointing opposite ways gives a negative one.
The angle formula
u · v equals |u||v|cos θ. Solving for cos θ turns the dot product into a way of measuring angles in any number of dimensions.
Zero means perpendicular
If the dot product is zero then cos θ is zero, so the angle is 90 degrees. This is the fastest perpendicularity test there is.
Why it matters
Work is force dotted with displacement. Only the part of the force along the motion counts, which is exactly what the dot product extracts.
Projection
The projection of u onto v is the shadow u casts on v. Its length is u · v divided by the magnitude of v.
A number measuring agreement
The dot product multiplies corresponding components and sums. The result is a scalar, and it is positive when the vectors point broadly the same way and negative when they oppose.
It encodes the angle
u · v = |u||v| cos θ, so the angle between any two vectors follows from their components. This works in any number of dimensions, where drawing the angle is impossible.
Zero means perpendicular
A dot product of zero with neither vector zero means the angle is 90°. It is the cleanest orthogonality test available, and it generalises to spaces you cannot picture.
Projection and work
The dot product measures how much of one vector lies along another, which is exactly what physics means by work. The definition was shaped by that application.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(4, 2) · length 4.47
- Vector b(-2, 4) · length 4.47
Step 3: Watch an Example
One step at a time.
Watch Owen Test Two Vectors
Owen checks whether u = ⟨3, −2, 1⟩ and v = ⟨2, 4, 2⟩ are perpendicular.
- Step 1
He multiplies the first components: 3 × 2 = 6.
Step 4: Your Turn
Practice makes it stick.
The Sled
Problem 1 of 2
A force ⟨6, 0, 0⟩ newtons moves a sled ⟨4, 3, 0⟩ metres. How much work is done, in joules?
The Right Angle
Problem 2 of 2
⟨2, k, 0⟩ is perpendicular to ⟨6, 3, 0⟩. What is k?
Dot Them Together
1 of 8
⟨1, 2, 3⟩ · ⟨4, 5, 6⟩. What is the result?
2 of 8
⟨1, 0, 0⟩ · ⟨0, 1, 0⟩. What is the result?
3 of 8
u · u where u = ⟨3, 4, 0⟩. What is the result?
4 of 8
Two vectors of length 5 and 4 meet at 60 degrees. What is the dot product?
5 of 8
A dot product is negative. Is the angle greater than 90 degrees? 1 yes, 0 no.
6 of 8
⟨5, 0, 0⟩ · ⟨−3, 8, 9⟩. What is the result?
7 of 8
Match each dot product value to what it says about the angle.
Tap a card on the left to start.
8 of 8
u and v both have length 2 and point the same way. What is u · v?
Step 5: Quick Check
Show what you know.
Question 1 of 2
⟨2, −1, 3⟩ · ⟨1, 4, 1⟩. What is the result?
Question 2 of 2
A dot product of zero tells you what?
What You Learned
- The dot product multiplies matching components and adds, giving a number.
- u · v = |u||v|cos θ, so it measures the angle between the vectors.
- A dot product of zero means the vectors are perpendicular.