AI Tutor
If you push a sled forward while a friend pulls it sideways, how much of your friend's effort actually helps the sled move forward?
The dot product is an operation that takes two directional vectors and multiplies their aligned parts to output a single real number (a scalar).
American physicist Josiah Willard Gibbs formalized modern vector notation and the dot product in the 1880s to simplify electromagnetic equations.
How do we calculate this single number when we only know the (x,y)(x, y) coordinates of our vectors?

Component Form Formula

By the algebraic definition, for vectors in component form uβƒ—=⟨u1,u2⟩\vec{u} = \langle u_1, u_2 \rangle and vβƒ—=⟨v1,v2⟩\vec{v} = \langle v_1, v_2 \rangle, the dot product is the sum of the products of their corresponding components.
u⃗⋅v⃗=u1v1+u2v2\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2
In 3D space, the rule extends naturally to three components: u⃗⋅v⃗=u1v1+u2v2+u3v3\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3.
Let's test this with a concrete 3D calculation.

Worked Example: 3D Vectors

Calculate aβƒ—β‹…bβƒ—\vec{a} \cdot \vec{b} for aβƒ—=⟨4,βˆ’2,5⟩\vec{a} = \langle 4, -2, 5 \rangle and bβƒ—=⟨3,6,1⟩\vec{b} = \langle 3, 6, 1 \rangle.
aβƒ—β‹…bβƒ—=(4)(3)+(βˆ’2)(6)+(5)(1)=12βˆ’12+5=5\vec{a} \cdot \vec{b} = (4)(3) + (-2)(6) + (5)(1) = 12 - 12 + 5 = 5