Dot Product Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Find ⟨1,2⟩⋅⟨3,−1⟩\langle 1, 2 \rangle \cdot \langle 3, -1 \rangle.

Solution

  1. 1
    Step 1: Multiply corresponding components and sum: 1(3)+2(−1)1(3) + 2(-1).
  2. 2
    Step 2: =3−2=1= 3 - 2 = 1.
  3. 3
    Note: The result is a scalar (number), not a vector.

Answer

11
The dot product multiplies corresponding components and sums the results, producing a scalar. A positive dot product means the vectors point in roughly the same direction (angle < 90°).

About Dot Product

The dot product of two vectors a=⟨a1,a2⟩\mathbf{a} = \langle a_1, a_2 \rangle and b=⟨b1,b2⟩\mathbf{b} = \langle b_1, b_2 \rangle is the scalar a⋅b=a1b1+a2b2\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, a⋅b=∥a∥∥b∥cos⁡θ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where θ\theta is the angle between the vectors.

Learn more about Dot Product →

More Dot Product Examples

Example 2 medium

Find the angle between [formula] and [formula].

Example 3 easy

Are [formula] and [formula] perpendicular?

Example 4 hard

Find [formula].

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