Advanced Properties of Vectors

Other then all the discussed properties, some other properties or operations on vectors are:

• Scalar Product or Dot Product
• Vector Product or Cross Product

Let’s discuss these in detail.

Dot Product

Dot Product, also known as the scalar product, is an algebraic operation that takes two vectors and returns a single scalar quantity. It is a measure of the extent to which two vectors are parallel and can be used to find the angle between the vectors.

For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product is defined as:

[Tex]\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_i b_i [/Tex]= |a||b| cos θ

Where,

• a_i and b_i are the components of vectors,
• a and b, respectively,
• n is the number of dimensions, and
• θ is the angle between both vectors.

Properties of Scalar Product of Vectors

Commutative Property: The dot product of two vectors is commutative i.e. a.b = b.a or |a| * b|cosθ = |b| * |a|cosθ.

Associative Property: The dot product is not associative in Nature or we can say that (a.b).c ≠ a.(b.c) , it is because (a.b).c represent a scalar (a.b ) multiplied to c vector whereas a.(b.c) represent a scalar (b.c) multiplied to a vector.

Distributive Property: The dot product of vectors is distributive over vector addition. So we can say that for three vector a , b and c

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

Orthogonality: Scalar product of two perpendicular vectors is 0 .Two vectors whose dot product is zero are known as orthogonal vectors.

Scalar product of two Parallel/Collinear/Like Vectors is the product of magnitude of the two vectors. Also the dot product of a vector with itself is the square of its magnitude i.e. a. a = |a|².

Cross Product

Cross product is the vector product of vectors. It represent the product of magnitude of the vectors and the sine of angle between them.

Unlike the scalar product give results a scalar value , Vector product always give a vector which is perpendicular to both the given vector. The vector product of two vectors a and b with an angle α between them is mathematically calculated as

a × b = |a| |b| sin α n̂

Where,

• ∣a∣ and ∣b∣ are the magnitudes of vectors a and b,
• α is the angle between the vectors, and
• n̂ is the unit vector in the direction of a × b.

Note: The direction of the resultant vector is determined by the right-hand rule.

Properties of Vector Product of Vectors

Commutative Property: The Cross product of two vectors isn’t commutative i.e. a × b ≠ b × a or |a| * b|sinθ ≠ |b| * |a|sinθ.

Associative property: The Vector product is Not associative in Nature or we can say that (a × b)×c ≠ a × (b × c) .

Distributive Property: The Cross product of vectors is distributive over vector addition. So we can say that for three vector a , b and c

a × (b+c) = a × b + a × c

Collinearity: Vector product of two Parallel / collinear /like vectors is 0 .

Vector product of two perpendicular vector is equal to product of the magnitude of the two vectors. Also the Vector product of a vector with itself is the square of its magnitude ie. a × a = |a|².

Vectors are one of the most important concepts in mathematics. Vectors are quantities that have both magnitude and direction. A vector quantity is represented by an arrow above its head. Vectors help us understand the behaviour of directional quantities in 2D and 3D planes. Vectors are also used for determining the position and change of position of points.

Every vector follows a certain set of rules, known as the properties of vectors. It is highly important to know these properties to have a strong command of vector algebra. In this article, we will see the definition of a vector, the properties of vectors, and the properties of vector products.