The dot product multiplies matching components and adds, giving a number. It measures agreement: u · v = |u||v| cos θ, so it delivers the angle between two vectors in any number of dimensions. Zero means perpendicular. The projection of u onto v is the shadow u casts on v.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What the number means
Same direction: large positive. Opposite: negative. Perpendicular: zero. ⟨1, 0, 0⟩ · ⟨1, 0, 0⟩ = 1; ⟨1, 0, 0⟩ · ⟨−1, 0, 0⟩ = −1; ⟨1, 0, 0⟩ · ⟨0, 1, 0⟩ = 0.
The angle formula
u = ⟨1, 2, 2⟩, v = ⟨2, 2, 1⟩: u · v = 2 + 4 + 2 = 8; |u| = |v| = 3. cos θ = 8/9, θ ≈ 27.3°. The dot product turns two lists of numbers into an angle.
Sign first
Positive, zero and negative already tell you acute, right or obtuse. ⟨1, 2, 2⟩ · ⟨−2, 1, 0⟩ = 0: right angle, no cosine needed. Check the sign before computing an angle.
A vector with itself
u · u = |u|², since cos 0 = 1. ⟨1, 2, 2⟩ · ⟨1, 2, 2⟩ = 1 + 4 + 4 = 9 = 3². A fast check on any magnitude.
Why it matters
Work is force dotted with displacement. A force ⟨3, 4, 0⟩ N moving an object through ⟨2, 0, 0⟩ m does 6 J: only the 3 N along the motion counts. A sideways force does no work.
Projection
The scalar projection of u onto v is u · v/|v|: the length of the shadow. ⟨3, 4, 0⟩ onto ⟨1, 0, 0⟩: 3/1 = 3. The vector projection is that length times the unit vector along v: ⟨3, 0, 0⟩.
Step 2: Try It Yourself
Tap and try it out.
u = ⟨1, 2, 2⟩, v = ⟨2, 2, 1⟩. Find the angle between them, and the projection of u onto v
- 1u · v = 2 + 4 + 2 = 8positive: acute
Step 0 of 4
- Vector a(4, 2) · length 4.47
- Vector b(-2, 4) · length 4.47
Step 3: In Real Life
Solar panels
A panel’s output is proportional to the dot product of the sun’s direction and the panel’s normal. Facing the sun, the dot product is largest. Installers tilt panels to maximize it.
Step 4: 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 components: 3 × 2 = 6, −2 × 4 = −8, 1 × 2 = 2.
Step 5: Your Turn
Practice makes it stick.
The Sled
Problem 1 of 2
A force ⟨6, 0, 0⟩ newtons moves a sled ⟨4, 3, 0⟩ meters. 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?
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 6: 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; the sign alone says acute, right or obtuse.
- A dot product of zero means the vectors are perpendicular, and u · u = |u|².
- The projection of u onto v has length u · v/|v|.