Vector Algebra: Definition, Properties, Operations and Examples

Vector algebra is an important topic in algebra. It revolves around the algebra of vector quantities. Scalars and vectors are two types of physical quantities, as we all know. While a scalar quantity comprises only magnitude, a vector quantity comprises both magnitude and direction.

Vector Algebra is an algebra in which the main elements are usually vectors. On vectors and vector spaces, we execute algebraic operations. This section contains rules and hypotheses based on vector properties and behaviour.

Let’s learn various concepts based on the basics of vector algebra including vector operations and properties with examples.

Properties of Vectors

The following properties of vectors help in better understanding vectors and are useful in performing numerous arithmetic operations involving vectors.

• Vector addition is both commutative and associative.

  I.e.,

  P + Q = Q + P (Commutative law)

  P + (Q + R)= (P + Q) + R (Associative law)

• Additive identity: For all P, there is a vector 0 such that (P + 0) = P = (0 + P). 
• Additive inverse: There is a vector -P for any vector P such that P + (-P) = 0. 
• Distributive (vector): r(P + Q) = rP + rQ 
• Associative (scalar): r(sP) = (rs)P 
• Multiplicative identity: 1P = P for any vector P 
• The dot product of two vectors is a scalar that is in their plane. 
• The cross product of two vectors is a vector that is perpendicular to the plane in which these two vectors are located.

Vector Algebra Operations

On vectors, we perform arithmetic operations like addition, subtraction, and multiplication, just like in regular Algebra. Vectors, on the other hand, have two terms for multiplication: dot product and cross product.

Addition of Vectors

Consider two vectors P and Q; the total of these two vectors can be computed when the tail of vector Q intersects the head of vector A. The size and direction of the vectors should not change during this addition. The vector addition obeys two key laws, which are as follows:

Commutative Law:

P + Q = Q + P

Associative Law:

P + (Q + R) = (P + Q) + R

Subtraction Of Vectors

The direction of the other vectors is inverted here, and the addition is performed on both vectors. If P and Q are the vectors for which the subtraction method must be used, we invert the direction of another vector, say Q, to make it -Q. We must now include vectors P and -Q. As a result, the vectors' directions are opposite, but their magnitude remains constant.

P – Q = P + (-Q)

Multiplication of Vectors

If k is a scalar quantity that is multiplied by a vector A, then kA represents the scalar multiplication. If k is positive, the vector kA will have the same direction as vector A, but if k is negative, the vector kA will have the opposite direction as vector A. The magnitude of vector kA is given by |kA|.

Dot Product

The dot product is otherwise also known as the scalar product. It is denoted by a dot (.) between two vectors. In this case, two equal-length coordinate vectors are multiplied to get a single value. A number or a scalar quantity is what we obtain when we take the scalar product of two vectors. The dot product of both vectors, if P and Q are two vectors, is given by;

P.Q = |P| |Q| cos θ

Cross Product

The multiplication symbol(x) between two vectors represents a cross product between them. It is a three-dimensional binary vector operation. If P and Q are two independent vectors, the outcome of their cross product (P x Q) is perpendicular to both vectors and normal to the plane containing both vectors. It is represented by;

P x Q = |P| |Q| sin θ