Section 2.3 - The Dot Product

Section 2.3 - The Dot Product

Introduction to the Dot Product

The dot product (also known as the inner product or scalar product) is a fundamental operation in vector algebra, commonly used in Calculus and Physics. It provides a way to multiply two vectors to obtain a scalar quantity, which has important geometric and analytic interpretations.

Definition of the Dot Product

• If u = (u1, u2, u3) and v = (v1, v2, v3), then the dot product of u and v is given by:

$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$

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

Properties of the Dot Product

• Commutativity: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$

• Distributivity over addition: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$

• Scalar multiplication: $(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (c\mathbf{v})$

• Dot product with itself: $\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$

• Zero vector: $\mathbf{u} \cdot \mathbf{0} = 0$

Geometric Interpretation

The dot product can also be defined in terms of the angle $\theta$ between two vectors u and v:

$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$

• If $\theta = 0^\circ$ (vectors point in the same direction), $\cos \theta = 1$ and the dot product is maximized.

• If $\theta = 90^\circ$ (vectors are perpendicular), $\cos \theta = 0$ and the dot product is zero.

• If $\theta = 180^\circ$ (vectors point in opposite directions), $\cos \theta = -1$ and the dot product is minimized (negative).

Examples

• Example 1: Find the inner product of $\mathbf{a} = (7, 0, 1)$ and $\mathbf{b} = (-2, 3, 5)$ and $5\mathbf{a} - (4\mathbf{b} - 2\mathbf{a})$. Additional info: This example involves both the computation of the dot product and the use of vector operations (scalar multiplication and addition/subtraction).

• Example 3: Find the angle between the vectors $\mathbf{a} = (7, 0, 1)$ and $\mathbf{b} = (-2, 3, 5)$. Additional info: Use the formula $\cos \theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$ to find the angle.

• Example 4: Find the angle between the vectors $\mathbf{a} = (3, 0, 1)$ and $\mathbf{b} = (0, -1, 0)$ and draw these two vectors on a Cartesian system. Additional info: Drawing vectors helps visualize their relative orientation and the angle between them.

• Example 6: Find the scalar projection of $\mathbf{b} = (1, 1, 2)$ onto $\mathbf{a} = (-2, 3, 1)$. Additional info: The scalar projection (also called the component) of $\mathbf{b}$ onto $\mathbf{a}$ is given by $\mathrm{comp}_{\mathbf{a}} \mathbf{b} = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}$.

Scalar Projection (Component of a Vector)

The scalar projection (or component) of vector u onto vector v is the length of the shadow of u in the direction of v. It is given by:

$\mathrm{comp}_{\mathbf{v}} \mathbf{u} = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|} = |\mathbf{u}| \cos \theta$

This formula is useful for finding how much of one vector lies in the direction of another.

Summary Table: Properties of the Dot Product

Property	Equation	Description
Commutativity	$\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$	Order does not matter
Distributivity	$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$	Distributes over vector addition
Scalar Multiplication	$(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})$	Scalars can be factored out
Self Dot Product	$\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$	Gives the square of the vector's magnitude
Zero Vector	$\mathbf{u} \cdot \mathbf{0} = 0$	Dot product with zero vector is zero

Applications of the Dot Product

• Determining the angle between two vectors

• Testing for orthogonality (perpendicularity): If $\mathbf{u} \cdot \mathbf{v} = 0$, then u and v are perpendicular.

• Finding projections and components of vectors

• Calculating work done by a force: $W = \mathbf{F} \cdot \mathbf{d}$, where F is force and d is displacement.