Dot Product Math Example 3

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

Example 3

easy
Are ⟨2,3⟩\langle 2, 3 \rangle and ⟨−3,2⟩\langle -3, 2 \rangle perpendicular?

Solution

  1. 1
    ⟨2,3⟩⋅⟨−3,2⟩=2(−3)+3(2)=−6+6=0\langle 2, 3 \rangle \cdot \langle -3, 2 \rangle = 2(-3) + 3(2) = -6 + 6 = 0.
  2. 2
    Since the dot product is 0, yes they are perpendicular.

Answer

Yes, they are perpendicular.
Two vectors are perpendicular (orthogonal) if and only if their dot product is zero. This gives a quick algebraic test for perpendicularity without computing angles.

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 1 easy

Find [formula].

Example 2 medium

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

Example 4 hard

Find [formula].

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