The dot product multiplies matching components and adds: (a₁, a₂) · (b₁, b₂) = a₁b₁ + a₂b₂. The result is a number, not a vector.
Step 1: Let's Learn
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The angle formula
The same product equals |a||b| cos θ. Rearranged, cos θ = (a · b) ÷ (|a||b|).
The perpendicular test
If the dot product is zero the vectors are perpendicular, because cos 90° = 0. This is the fastest right-angle test there is.
What the sign says
A positive dot product means the vectors point broadly the same way; negative means broadly opposite.
Where it is used
Work in physics is force dotted with displacement. Pushing sideways to your motion does no work, which is the zero case again.
Two vectors in, one number out
The dot product multiplies corresponding components and adds: ⟨a, b⟩ · ⟨c, d⟩ = ac + bd. The result is a scalar, not a vector, which is why it is also called the scalar product.
It encodes the angle
u · v = |u||v| cos θ. Rearranged, this gives the angle between any two vectors from their components alone. It is the standard way to compute an angle in two, three or any number of dimensions.
Zero means perpendicular
If the dot product is zero and neither vector is zero, cos θ is zero and the vectors are at right angles. This is the cleanest perpendicularity test there is, and it works in every dimension.
Projection and work
The dot product measures how much of one vector points along another, which is exactly what physics means by work: force times distance in the direction of motion. The definition was built for that purpose.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 1)
- Vector b(-1, 3)
Step 3: Watch an Example
One step at a time.
Watch Elena Test Two Vectors
Elena wants to know whether (3, 1) and (−1, 3) meet at a right angle.
- Step 1
She multiplies the x components: 3 × −1 = −3.
Step 4: Your Turn
Practice makes it stick.
The Product
Problem 1 of 2
(2, 5) · (4, 1). What is the result?
The Right Angle
Problem 2 of 2
(6, 2) · (−1, 3). What is the result?
Dot Them
1 of 8
(1, 2) · (3, 4)?
2 of 8
(5, 0) · (0, 7)?
3 of 8
(2, 3) · (2, 3)?
4 of 8
(−2, 1) · (4, 2)?
5 of 8
Magnitude of (3, 4)?
6 of 8
a · b = 0. What is the angle between them, in degrees?
7 of 8
Sort each pair by what the sign of its dot product tells you.
Tap something to move it.
- Empty
- Empty
8 of 8
(4, 0) · (0, 9)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
(3, 2) · (1, 4)?
Question 2 of 2
What does a dot product of zero mean?
What You Learned
- The dot product multiplies matching components and adds, giving a number.
- It equals |a||b| cos θ, so it delivers the angle between two vectors.
- A dot product of zero means perpendicular.