**Properties of Scalar Product or Dot Product :**

Here we are going to see some properties of scalar product or dot product.

**Property 1 :**

Scalar product of two vectors is commutative.

With usual definition,

a vector ⋅ b vector = |a||b|cos θ = |b||a|cos θ = b ⋅ a

That is, for any two vectors a and b, a ⋅ b = b ⋅ a

**Property 2 :**

Nature of scalar product

We know that 0 ≤ θ ≤ π

If θ = 0 then a ⋅ b = ab

[Two vectors are parallel in the same direction then θ = 0]

If θ = π then a ⋅ b = −ab

[Two vectors are parallel in the opposite direction θ = π/2

- If θ = π/2 then a vector ⋅ b vector[Two vectors are perpendicular θ = π/2].
- If 0 < θ < π/2 then cosθ is positive and hence a ⋅ b is positive.
- If π/2 < θ < π then cos is negative and hence a ⋅ b is negative.

That is a vector ⋅ b vector is

**Property 3 :**

When is a scalar/dot product of two vectors equal to zero ?

a vector ⋅ b vector = 0

when |a vector| = 0 |(or) |b vector| = 0 or θ = π/2

**Property 4 :**

If the dot product of two nonzero vectors is zero, then the vectors are perpendicular.

For any two non-zero vectors a vector and b vector, a ⋅ b = 0 a vector is perpendicular to b vector.

**Property 5 :**

Different ways of representations of a vector ⋅ b vector

a vector⋅a vector =|a vector|^{2} = (a vector)^{2} = (a vector)^{2} = a^{2} .

These representations are essential while solving problems

**Property 6 :**

For any two scalars λ and μ

λa vector ⋅ μb vector = λμ (a vector ⋅ b vector) = (λμa vector) ⋅ b vector = a vector ⋅ (λμb vector)

**Property 8 :**

Scalar product is distributive over vector addition.

That is, for any three vectors a,b,c

a vector (b vector + c vector) = a ⋅ b + a ⋅ c (Left distributivity)

(a vector + b vector) ⋅ c vector = a ⋅ c + b ⋅ c (Right distributivity)

Subsequently,

a vector ⋅ (b vector − c vector) = a vector ⋅ b vector - a vector ⋅ c vector

and (a vector − b vector) ⋅ c vector = a vector ⋅ c vector − b vector ⋅ c vector

These can be extended to any number of vectors

**Property 9 :**

Vector identities

**Property 10 :**

Working rule to find scalar product

of two vectors

Let

Hence, the scalar product of two vectors is equal to the sum of the products of their corresponding rectangular components.

**Property 11 :**

Angle between two vectors

**Property 12 :**

For any two vectors and a vector b vector

|a vector + b vector| ≤ |a vector| + |b vector|

We know that if a vector and b vector are the two sides of a triangle then the sum a vector + b vector represents the third side of the triangle. Therefore, by triangular property, |a vector + b vector| ≤ |a vector| + |b vector|

**Property 13 :**

For any two vectors and, |a vector ⋅ b vector| ≤ |a vector| |b vector|.

If one of them is zero vector then the equality holds. So, let us assume that both are non-zero vectors.

After having gone through the stuff given above, we hope that the students would have understood,"Properties of Scalar Product or Dot Product"

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