Vector Algebra - Vector Algebra

Vector algebra is an important topic of the Class XII Maths subject as it deals with symbols, letters, numbers, and variables. These concepts are later used in equations, formulae, and expressions. In vector algebra, the elements used are denoted as vectors. A vector algebra has some rules and the algebraic operations are performed on the vectors. The vector algebra section carries a weightage of around 14 marks in the examination. 

Types of Vectors:

Below mentioned are the different types of vectors:

• Zero Vector
• Unit Vector
• Coinitial Vector
• Equal Vectors
• Negative of A Vector

What is the Definition of Vector?

A vector is an object that has direction and magnitude. An arrow is used for representing the direction and length shows the magnitude. Arrow is on the head of the vector, V, and the magnitude is represented as │V│. Any two vectors whose magnitudes and directions are equal are said to be equal. The arithmetic operations such as addition, multiplication, and subtraction can be performed on vectors.

Addition of Vectors – It has two basic laws. In the following example, there are two vectors P and Q:

Associative Law: P+(Q+R) = (P+Q)+R

Commutative Law: P+Q = Q+P

Multiplication of Vectors – In an equation if k is a scalar quantity and A is the vector, then the multiplication is represented as kA.

Subtraction of Vectors – P–Q = P+(-Q), in this formula the direction of the vectors is reversed but the magnitude remains the same.

A vector uses two terminologies which are a dot product and a cross product.

Dot Product – It is represented by a (.) between two vectors, and is also known as the scalar product. In this method, the two coordinates are of equal length and are multiplied to get a

single number. If P and Q are the two vectors then the dot product can be represented as:

P.Q = │P││Q cosƟ

Cross Product – Across product between two products is denoted by ‘x’. If P and Q are the two vectors, then the cross product is represented as below:

P x Q = │P││Q│sinƟ

1. Calculate the dot product of P (1, 3, -4) and Q (3, -5, 2) vectors?

Solution.

As per the dot product formula: P.Q = P1Q1 + P2Q2 + P3Q3 + and so on

Thus, P.Q = 1 . 3 + 3 . (-5) + (-4) . 2

P.Q = 3 – 15 – 8

P.Q = -20

2. Classify the following concepts as scalar or vectors?

Time

Distance

Force

Solution. 

Time: It is a scalar, as it has only magnitude and no direction. For instance, 5 seconds.

Distance: It is a scalar, as it has only magnitude and no direction. For instance, 5 meters.

Force: It is a vector, as it has a magnitude and direction. For instance, 5 N.

3. Classify the following in scalars and vectors?

10 kg

40 watt

2 metres North-West

Solution.

10 kg is a scalar quantity as it has only magnitude.

40 watt is a scalar quantity as it has only magnitude.

2 metres North-West is a vector quantity as it has both direction and magnitude.

Q: What are the two types of vector multiplication?

Q: Which are the important concepts of Vector Algebra?

Q: What are the different types of vectors?

• Zero Vector
• Unit Vector
• Negative Vector
• Collinear Vector
• Co-initial Vector
• Equal Vector

Q: Which are the two types of physical quantities used in vector algebra?

Q: Which are the properties of vector addition?