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Dot Product Calculator – Free Online Calculator

Dot Product Calculator

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Dot Product Calculator

Definition of Dot Product

The dot product, also celebrated as the scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This procedure is defined as the sum of the products of the corresponding entries of the two sequences of numbers. Mathematically, the dot product of two vectors a = [a1, a2, …, an] and b = [b1, b2, …, bn] is as follows:

\[ \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 + \ldots + a_n \times b_n \]
\[ \text{Or, } \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i \]

Example: Consider two vectors a = (1, 2, 3) and b = (4, -5, 6). Their dot product is calculated as: a.b

\( \mathbf{a} \cdot \mathbf{b} = (1 \times 4) + (2 \times -5) + (3 \times 6) \)

\( = 4 – 10 + 18 \)

\( = 12 \)

The dot product can take on positive real numbers, negative real numbers, or zero as its values.

Matrix Representation of Dot Product

This will yield the sum of the products of conforming components from both vectors.

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Properties of Dot Product

The dot product claims several fundamental properties that focus its mathematical utility:

  1. Commutative Property: a⋅b = b⋅a 

Example: If a = (1, 3) and b = (4, 2), then a·b = b·a = 4 + 6 = 10.

  1. Distributive Property: For any three vectors a, b, and c, the scalar product distributes over both addition and subtraction. This characteristic extends to any set of vectors and allows the scalar product property to apply to multiple vectors. 

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Example: If a = (1, 2), b = (2, 1), and c = (0, 1), then 

a·(b + c) = (1, 2)·(2 + 0, 1 + 1) = 4.

  1. Scalar Multiplication (Natural Property): c(a⋅b) = (ca)⋅b = a⋅(cb) 

Example: If a = (1, 2), b = (3, 4), and c = 2, then 

2(a·b) = 2(1×3 + 2×4) = 2(11) = 22.General Properties: Zero Vector: a·0 = 0

Nature of Dot Product

Considering the nature of the dot product, where 0 ≤ θ ≤ π:

  1. If θ = 0, the dot product (a ⋅ b) equals ab which indicates the two vectors are parallel in the same direction.
  2. If θ = π, the dot product becomes -ab, which signifies the vectors are parallel but in opposite directions.
  3. When θ = π/2, the dot product is 0 which indicates the vectors are perpendicular.
  4. For 0 < θ < π/2, cosθ is positive which results in a positive dot product.
  5. If π/2 < θ < π, cosθ is negative which leads to a negative dot product.

Exploring additional properties of the dot product:

  1. When λ is a scalar, the dot product of (λa).b = λ(a.b)
  2. For any scalars λ and μ, λa . μb = (λμ a).b = a.(λμ b).
  3. The length of a vector (|→a|) is the square root of its dot product with itself: |a| = √(a⋅a).
  4. a⋅a = |a|² or can be expressed as a²; where |a| is the magnitude of a.
  5. For any vectors a and b, the magnitude of their sum (|a + b|) is always less than or equal to the sum of their individual magnitudes: |a + b| ≤ |a| + |b|.

Guideline to Use the Dot Product Calculator

Using the Dot Product Calculator is as simple as you can’t think of. Just do the following:

For example: a = (2, 2, 3) and b = (4, -5, 6)

  1. Input vectors per line. Such as:

Input 2, 2, 3 then press Enter. Then 

Write 4, -5, 6

  1. Each component will be joined by comma.
  2. Press CALCULATE. You will find your answer in the Answer Section with step-by-step solution.
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