Dot Product of Two Vectors

This tutorial explains the dot product (scalar product) of two vectors, its properties, geometric meaning, and applications such as finding angles and determining whether vectors are orthogonal.

Definition of the Dot Product

Let \[ \mathbf{v} = \langle v_1, v_2 \rangle \quad \text{and} \quad \mathbf{u} = \langle u_1, u_2 \rangle \] be two vectors in the plane. The dot product of v and u is defined by:

\[ \mathbf{v} \cdot \mathbf{u} = v_1u_1 + v_2u_2 \]

Important: The result of a dot product is a scalar (a real number), not a vector.

Example 1: Computing a Dot Product

Given: \[ \mathbf{v} = \langle -2, 3 \rangle, \quad \mathbf{u} = \langle 4, 6 \rangle \] find \( \mathbf{v} \cdot \mathbf{u} \).

Solution

Apply the definition of the dot product:

\[ \mathbf{v} \cdot \mathbf{u} = (-2)(4) + (3)(6) \] \[ = -8 + 18 = 10 \]

The dot product of the two vectors is 10.

Properties of the Dot Product

Example 2: Vector Magnitude Using the Dot Product

Let: \[ \mathbf{v} = \langle 3, -4 \rangle \]

Solution

Method 1: Definition

\[ \|\mathbf{v}\| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \]

Method 2: Dot Product Property

\[ \mathbf{v} \cdot \mathbf{v} = 3^2 + (-4)^2 = 25 \] \[ \|\mathbf{v}\| = \sqrt{25} = 5 \]

Both methods give the same magnitude, confirming the property.

Geometric Interpretation of the Dot Product

If \( \theta \) is the angle between vectors \( \mathbf{v} \) and \( \mathbf{u} \), then:

\[ \mathbf{v} \cdot \mathbf{u} = \|\mathbf{v}\| \, \|\mathbf{u}\| \cos \theta \] Dot product and cosine rule

This formula allows us to compute the angle between two vectors and determine whether they are orthogonal.

If \( \cos \theta = 0 \), then \( \theta = \frac{\pi}{2} \) and the vectors are perpendicular.

Example 3: Orthogonal Vectors

Show that: \[ \mathbf{v} = \langle 3, -4 \rangle, \quad \mathbf{u} = \langle 4, 3 \rangle \] are orthogonal.

Solution

\[ \mathbf{v} \cdot \mathbf{u} = (3)(4) + (-4)(3) = 12 - 12 = 0 \]

Since the dot product is zero, the vectors are orthogonal.

Example 4: Angle Between Two Vectors

Find the angle between: \[ \mathbf{v} = \langle 1, 1 \rangle, \quad \mathbf{u} = \langle -4, 3 \rangle \]

Solution

\[ \mathbf{v} \cdot \mathbf{u} = (1)(-4) + (1)(3) = -1 \] \[ \|\mathbf{v}\| = \sqrt{2}, \quad \|\mathbf{u}\| = 5 \] \[ \cos \theta = \frac{-1}{5\sqrt{2}} \] \[ \theta = \cos^{-1}\!\left(\frac{-1}{5\sqrt{2}}\right) \approx 98.1^\circ \]

Exercises

  1. Given \( \mathbf{v} = \langle 10, -5 \rangle \) and \( \mathbf{u} = \langle 2, u_2 \rangle \), find \( u_2 \) so the vectors are orthogonal.
  2. Find the angle between \( \mathbf{v} = \langle 1, 1 \rangle \) and \( \mathbf{u} = \langle -2, -2 \rangle \).

Answers

  1. \( u_2 = 4 \)
  2. \( 180^\circ \)

More Resources

Vector Calculators
Vector Tutorials