Dot Product of Two Vectors

June 1, 2023

Dot Product of Two Vectors

The dot product, called the scalar product, is a binary operation between two vectors resulting in a scalar quantity. Algebraically, it is computed as the sum of the products of the corresponding entries of two sequences of numbers.

Geometrically, it can be understood as the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

The dot product is a fundamental concept in linear algebra and finds numerous applications in fields such as geometry, mechanics, engineering, and astronomy. 

It is used to calculate angles between vectors, project one vector onto another, find the magnitude of a vector, determine whether two vectors are orthogonal, and solve systems of linear equations.

What is a Dot Product

A vector is a mathematical object that possesses both magnitude and direction. Vectors can be subjected to various mathematical operations, including addition and multiplication. The multiplication of vectors can be achieved through two distinct methods: the dot product and the cross product.

In this article, we will focus on the dot product of two vectors and illustrate its properties and applications through a series of examples.

Properties of Dot Product

The properties of the dot product of vectors are:

  1. Commutative property
  2. Distributive property
  3. Bilinear property
  4. Scalar Multiplication property
  5. Non-Associative Property
  6. Orthogonal Property

Commutative Property

The dot product of two vectors is commutative, meaning that a.b is equal to \(b . a\). This property allows for greater flexibility when performing vector operations.

\( a . b = b . a \)

\( a . b = |a| |b| cosθ \)

\( a . b = |b| |a| cosθ \)

Distributive Property

The dot product is distributive, meaning that a vector dotted with the sum of two other vectors equals the sum of the dot products of the original vector with each of the other vectors.

\( a . (b + c) = a . b + a . c \)

Bilinear Property

The dot product is bilinear, meaning it is linear in both arguments. In other words, scaling one of the vectors or adding another vector will scale or add the dot product accordingly. 

\( a . (rb + c) = r . (a . b) + (a . c) \)

Scalar Multiplication Property

The dot product satisfies the scalar multiplication property, meaning that scaling both vectors by the same scalar factor is equivalent to scaling the dot product by the product of the same scalar factor. 

\( (xa) . (yb) = xy (a . b) \)

Non-Associative Property

The dot product is non-associative, meaning that the order in which we perform the dot products of three vectors matters. In other words, \( (a . b) . c \) is not necessarily equal to \( a . (b . c)\). 

Orthogonal Property

Two vectors are orthogonal only when \( a . b = 0 \)

Example: Calculate the dot product of vectors a and b: a = [2, 3, -4] and b = [-1, 5, 2]

We can use the formula : \( a . b = ax × bx + ay × by + az × bz \)

\( a . b = 2 × (-1) + 3 × 5 + (-4) × 2 \)

\( a . b = -2 + 15 – 8 \)

\( a . b = 5 \)

The dot product of vectors a and b is 5.

Example 2: Find the dot product of two vectors, a and b, where |a| = 3 and |b| = 5, and the angle θ between them is 45 degrees. 

Using the formula \( a . b = |a| |b| cosθ \), we have:

\( a . b = 3 × 5 × cos45° \)

Recall that \( cos45° = \dfrac{\sqrt{2}}{2} \), so we can substitute that in:

\( a . b = 3 × 5 × \dfrac{\sqrt{2}}{2} \)

Simplifying, we get:

\( a . b = 15 × \dfrac{\sqrt{2}}{2} \)

Finally, we can approximate this value to a decimal by multiplying and simplifying:

\( a.b ≈ 10.61 \)

Therefore, the dot product of vectors a and b is approximately 10.61.

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