Vector algebra

Vector algebra is one of the essential topics of algebra. It studies the algebra of vector quantities. As we know, there are two types of physical quantities, scalars and vectors. The scalar quantity has only magnitude, whereas the vector quantity has both magnitude and direction. Learn about Magnitude Of A Vector here.

Algebra is a significant subject in Maths where we use universal symbols or letters to signify the quantities, numbers and variables. These symbols are later used in many expressions, equations and formulae, to perform algebraic operations. It has many branches.

In essence, vector algebra is an algebra where the essential elements usually denote vectors. We perform algebraic operations on vectors and vector spaces. This branch has rules and hypotheses based on the properties and behaviour of vectors. Here, you will learn various concepts based on the basics of vector algebra and some solved examples.

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Definition

A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail.

Vector Algebra Operations

Just like in usual Algebra, we also perform arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.

Addition of Vectors

Let us consider there are two vectors P and Q, then the sum of these two vectors can be performed when the tail of vector Q meets with the head of vector A. And during this addition, the magnitude and direction of the vectors should not change. The vector addition follows two important laws, which are;

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

Subtraction Of Vectors

Here, the direction of other vectors is reversed and then the addition is performed on both the given vectors. If P and Q are the vectors, for which the subtraction method has to be performed, then we invert the direction of another vector say for Q, make it -Q. Now, we need to add vector P and -Q. Thus, the direction of the vectors are opposite each other, but the magnitude remains the same.

• P – Q = P + (-Q)

Multiplication of Vectors

If k is a scalar quantity and it is multiplied by a vector A, then the scalar multiplication is given by kA. If k is positive then the direction of the vector kA is the same as vector A, but if the value of k is negative, then the direction of vector kA will be opposite to the direction of vector A. And the magnitude of the vector kA is given by |kA|.

Dot Product

The dot product is often called a scalar product. It is represented using a dot(.) between two vectors. Here, two coordinate vectors of equal length are multiplied in such a way that they result in a single number. So basically when we take the scalar product of two vectors, the result is either a number of a scalar quantity. Suppose P and Q are two vectors, then the dot product for both the vectors is given by;

• P.Q = |P| |Q| cos θ

If P and Q are both in the same direction, i.e. θ = 0°, then;

• P.Q = |P| |Q|

If P and Q are both orthogonal, i.e. θ = 90°, then;

• P.Q = 0 [since cos 90° = 0]

In vector algebra, if two vectors are given as;

P = [P₁,P₂,P₃,P₄,….,P_n] and Q = [Q₁,Q₂,Q₃,Q₄,….,Q_n]

Then their dot product is given by;

• P.Q = P₁Q₁+P₂Q₂+P₃Q₃+……….P_nQ_n

Cross Product

A cross product is denoted by the multiplication sign(x) between two vectors. It is a binary vector operation, defined in a three-dimensional system. If P and Q are two independent vectors, then the result of the cross product of these two vectors (P x Q) is perpendicular to both the vectors and normal to the plane that contains both the vectors. It is represented by;

• P x Q = |P| |Q| sin θ

Vector Algebra Formulas

Apart from the addition, subtraction and multiplication, there are some other formulas of vectors in algebra. They are:

• \(\begin{array}{l}\text{Let P(x, y, z) be a point. The position vector of P is } \vec{OP}=\vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \\ \text{and the magnitude of this vector is given by } |\vec{OP}|=|\vec{r}|=\sqrt{x^2+y^2+z^2}\end{array} \)
  .
• Suppose for any vector, r is the magnitude, (l, m, n) are the direction cosines and (a, b, c) are the direction ratios, then:
  l = a/r, m = b/r, n = c/r
• \(\begin{array}{l}\text{Let } |\vec{a}| \text{ be any vector, then the unit vector in the direction of }|\vec{a}| \text{ is given by } \hat{a}=\frac{\vec{a}}{|\vec{a}|}\end{array} \)
• \(\begin{array}{l}\text{The position vector of a point P dividing a line segment joining the points A and B whose} \\ \text{ position vectors are }|\vec{a}| \text{ and }|\vec{b}| \text{respectively, in the ratio m : n internally is given by } \frac{n\vec{a}+m\vec{b}}{m+n}\end{array} \)
• \(\begin{array}{l}\text{In the case of external division, the formula becomes: } \frac{m\vec{b}-n\vec{a}}{m-n}\end{array} \)
• \(\begin{array}{l}\text{Matrix representation of the cross product of two vectors is given by: } \vec{a}\times\vec{b}=\begin{vmatrix} \hat{i} &\hat{j} &\hat{k} \\ a_1 &b_1 &c_1 \\ a_2 &b_2 & c_2 \end{vmatrix}\end{array} \)

Vector Algebra Example

Question: Find the dot product of vectors P(1, 3, -4) and Q(3,−5, 2).

Ans: As per the vector algebra definition of the dot product, we know;

P.Q = P₁Q₁+P₂Q₂+P₃Q₃+……….P_nQ_n

Therefore,

P.Q = 1.3 + 3.(-5) + (-4).2

= 3 – 15 – 8

= -20