Vector Product: Overview, Questions, Preparation

A vector is an entity that has a magnitude as well as direction. The length represents the magnitude of the vector and the arrow represents the direction. The two ways of multiplying vectors are as follows:

• A Dot product is also known as the scalar product of vectors.
• Cross product is also called a vector product.

Vector is an important concept in Algebra and carries a weightage of around 14 marks in the examinations. Learning this concept would help you to score well in your examinations.

What is a Vector Product?

A Cross product is a binary operation of two vectors that results in a vector that is perpendicular to both vectors. The cross product of two vectors is calculated with the use of the right-hand rule. A right-hand rule is nothing but that the resultant of any two vectors is perpendicular to the other given vectors. With the use of vector products, we can also calculate the magnitude of the remaining vector.

Vector product of A and B vectors is denoted as AxB

The formula for Cross Product (Vector Product) is:

AxB = AB sinƟ
Cross Product of Two Vectors
XxY = │X│.│Y│sinƟ
Cross Product of Three Vectors (A, B, and C are the vectors)
A x (BxC) = (A.C) B – (A.B) C
(AxB) x C = -C x (AxB) - -(C.B) A + (C.A) B

What are the properties of Vector Products?

• Anti-Commutative Property - Vector product do not have commutative property i.e. axb = - (bxa)
• Jacobi Property - In case of vector multiplication (ka) x b = k (bxa) = a x (kb) would hold true
• Zero vector property - If the vectors are collinear then, axb = 0

In terms of unit vectors axb can be represented as:

a = a1 i^ + a2 j^ + a3 k^

b = b1 i^ + b2 j^ + b3 k^

Distributive law – a x b (b+c) = a x b + a x c

With these properties we can understand how to calculate the vector product of given vectors.

1. A = i + 2j + 3k, B = 2i + 3j – 2k, C = i + j + k

Calculate A x (BxC), if A x (BxC) = (A.C) B – (A.B) C

Solution.

A.C = (i+2j+3k) . (i+j+k) = 6

A.B = (i+2j+3k) . (2i+3j–2k) = 2

Then, A x (BxC) = 6 (2i+3j–2k) – 2 (i+j+k)
= 10i - 16j – 14k

2. Calculate a vector product, where A = (2, 3, 4) and B = (5, 6, 7).

Solution.

In this, A_x = 2, A_y = 3, and A_z = 4

B_x = 5, B_y = 6, and B_z = 7

Thus, putting the values in the formula:

C_x = A_y . B_z – A_z . B_y

C_y = A_z . B_x – A_x . B_z

C_z = A_x . B_y – A_y . B_x

we would get C_x = -3, C_y = 6, and C_z = 3

3. If there are two vectors lal = 4 and lbl = 2 and Ө = 60⁰. Calculate the dot product

Solution.

a.b = lal lbl cos Ө

a.b = 4.2 cos 60⁰

a.b = 4.2 x (½)

a.b = 4

Q: When there are two vectors, then is A x B equal to B x A?

Q: What is the formula for calculating a vector product?

Q: Which rule is used for calculating a vector product?

Q: What are the properties of a vector product?

Q: Which are the two ways of multiplying a vector?