Dot Product

Definition

The dot product, also known as the scalar product or inner product, is an operation that takes two vectors and produces a scalar quantity. Given two vectors a and b in n-dimensional space, the dot product is defined as:

$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+\dots +a_n{b}_n$

where $a_1,a_2,\dots ,a_n$ are the components of vector a and $b_1,b_2,\dots ,b_n$ are the components of vector b.

Another way to express the dot product is:

$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}|\cdot |\mathbf{b}|\cdot \cos (\theta )$

where $|\mathbf{a}|$ and $|\mathbf{b}|$ are the magnitudes of vectors a and b respectively, and $\theta$ is the angle between the two vectors.

Properties

• Commutativity: $\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a}$

• Distributivity: $\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c}$

• Scalar Multiplication: $(c\mathbf{a})\cdot \mathbf{b}=c(\mathbf{a}\cdot \mathbf{b})$

• Orthogonality: If $\mathbf{a}\cdot \mathbf{b}=0$, then the vectors a and b are orthogonal (perpendicular).

Applications

The dot product is widely used in various areas of mathematics and physics. In geometry, it can be used to determine the angle between two vectors, calculate projections, find orthogonal projections, and much more. In physics, the dot product is used in mechanics to calculate work done, in electricity and magnetism for calculating electric work, and in optics for calculations related to light.

Real-life Example

Consider a force F acting on an object and displacing it by a distance d. The work done by the force can be calculated using the dot product:

$W=\mathbf{F}\cdot \mathbf{d}$

If the force and displacement are in the same direction, the work done is positive. If they are in opposite directions, the work done is negative.

Historical Context

The dot product was introduced by Hermann Grassmann, a German mathematician, in the 19th century. It was further developed and popularized by Josiah Willard Gibbs and Oliver Heaviside. The dot product plays a fundamental role in vector algebra and calculus, providing a way to multiply vectors and relate them geometrically.

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Exam Questions

1. Calculate the dot product of vectors a = (2, -1, 3) and b = (4, 0, -2).
2. Prove that the dot product of two parallel vectors is equal to the product of their magnitudes.
3. Two forces of 10 N and 15 N act on an object at an angle of 60 degrees to each other. Find the work done by the forces if the object is displaced by 5 meters.

Enjoy solving these problems and deepening your understanding of the dot product!