Vector Algebra

What Is Meant By Vector?

Physical quantities are of two types: scalar and vector.

Scalar Quantities: They are the quantities which have only magnitude and no direction. Example: mass, distance, time, speed, volume, density, pressure, work, etc.

Vector Quantities: They are the quantities which have both magnitude and direction. 

Example: displacement, velocity, acceleration, force, weight, momentum, etc.

Representation of Vector Quantities

A vector quantity is represented by an arrow. This arrow is called the ‘vector’. The length of the arrow represents the magnitude and the head of the arrow represents direction.    

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Suppose a car A is running with a speed of 10m/s towards the east and another car B running with 20m/s towards the northeast. These velocities can be represented by the vector shown below:

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The vector quantities can be written by putting arrowsover the bold letter.  If the magnitude of a vector quantity is A then it is written as A

Equality of Vector Algebra

Two or more vectors are said to be equal if, and only if, they have the same magnitude and same direction. In fig 3 PQ, RS, AB, TU and XY are equal vectors. If the direction of the vectors is reversed, the sign of the vector is reversed. We can call it the “negative vector” of the original vector. 

In fig 3, vector AB is a negative vector of all the other vectors.

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Operations on Vector Algebra

The operations like addition, subtraction, multiplication, and division are possible on algebra. The operation on algebra is more like arithmetic operations however, in the case of multiplication the product cannot be found out by simple algebraic method but by dot and cross product. Some of the vector calculations rules are given below:

  • Addition of Vectors

Since vectors have both magnitude and direction, they cannot be added by ordinary algebra; instead we can add the magnitude of the two vectors by simple algebra. 

Let \[\underline A \] and \[\underline B \]be the two vectors and \[\underline R \] be the sum of the vectors. 

                         \[\underline R  = \underline A  + \underline B \]

The magnitude of \[\underline A  + \underline B \] can be determined by the length of R whereas the direction can be determined by measuring the angle between \[\underline R \]  and  \[\underline A \] or  \[\underline B \]. Incase we exchange the position of  A and  B(shuffle) in the formula then \[\underline R \] can be written as \[\underline {R'} \]such as: 

                         \[\underline {R'}  = \underline B  + \underline A \]


Methods of Vector Addition:

  1. Triangle Law of Vector Addition: This states that if two vectors are represented in magnitude and direction by two sides of the triangle, then the resultant of the triangle taken in the opposite order is represented by the third side.

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 Let two vectors P and Q represent the magnitude and direction by the sides OA and AB.

Then, the resultant R will be represented by,

 Example 1: Given that , find the sum of the vectors.

 Solution: According to the triangle law of vector addition, the sum of vector PQ and QR is equal to vector PR. 

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       On adding the two vectors, we get 

 

  1. Parallelogram of Vector Addition:  According to this law states that if two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonals of the parallelogram drawn from the same point.

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Let two vectors  A and Binclined to each other at an angle represented in magnitude and direction by the sides of QR and QP. Then according to a parallelogram, the resultant of the given two vectors will be represented by the diagonal of the parallelogram.

Note: The resultant of two vectors is maximum when they are in the same direction, and is minimum when they are in the opposite direction. Further in parallelogram if one diagonal is the sum of the sides then the other diagonal is the sum of the differences.


Example 2: By using the parallelogram law of vector addition, prove that vector addition is commutative.

Solution: Let us consider a parallelogram of length a and breadth of length b. Then according to the law of parallelogram, the resultant vector would be the diagonal of the parallelogram.

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Thus, the diagonal is equal to a+b as well as b+a. This means a+b = b+a. 


  • Subtraction of Vectors

Same as the addition of vectors, the algebra rule is followed while subtracting one vector from another. The subtraction of two vectors can be done by reversing the direction of the second vector and then adding them together.

For example: If A and B are the two vectors, then we can simply add A and B by inverting the direction of B by -B. The resultant vector R will be,


R = A - B = A + (-B).

R’ = B - A.


Example 3: 

Solution: 

  • Multiplication of Vectors

On multiplying a vector by a scalar number (suppose k), the results can be given by kA. The magnitude of R is k times the magnitude of A and the direction of R is the same as that of A

R = kA.

There are two types of vector multiplication:

  1. Dot Product

The scalar product of two vectors is defined as a scalar quantity equal to the product of their magnitude and the cosine of the angle between them. Suppose A and B are two vectors, then the dot product for both the vectors is given by;

  • A.B = |A| |B| cos θ

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

  • A.B = |A| |B|

If A and B are both orthogonal, i.e. θ = 90°, then;

  • A.B = 0 [since cos 90° = 0]


Example 4: If A = -2i + 2j -k and B = 3i + 6j +2k. Find the angle between A nd B.

Solution: According to the vector algebra formulas for dot product, 

\[A.B = \left| A \right|\left| B \right|cos\theta \]


\[\cos \theta = \frac{{AB}}{{AB}} = \frac{4}{{\left( 3 \right)\left( 7 \right)}} = 0.19\]

= 79 degrees.


  1. Cross Product

The vector product of two vectors is defined as a vector having magnitude equal to the product of magnitudes of the two vectors and the sine of the angles between them. Also, having the direction perpendicular to the plane containing the two vectors.

If A and B are two independent vectors, then the result of the cross product of A and B that is, (A x B) will be perpendicular to both the vectors and normal to the plane that contains both the vectors. It is represented by;

  • A x B = |A| |B| sin θ


Example 4: If A = 3i + j + 2k and B = 2i - 2j +4k. Find the A x B.

Solution: According to vector rules for cross product. The Determinant of AxB is equal to the resultant of the cross product of A and B that is 8i - 8j - 8k.

FAQ (Frequently Asked Questions)

Question 1: Can You Associate Vectors with (a) the Length of a Wire Bent into a Loop (b) a Plane Area, ( c ) a Sphere? Explain.

Answer: a) No; a vector cannot be associated with the length of a bent wire, because at every point of a bent wire the direction is changing.

b) Yes, a vector can be associated with a plane area and its direction is along the normal to the plane, drawn outward.

c) No, a vector cannot be associated with a sphere, because the direction of the normal drawn outward on the surface of the sphere is changing at every point. 

Question 2: When is the Sum of Two Vectors Maximum and When Minimum?

Answer: The sum of two vectors maximum when both are in the same direction and minimum when in the opposite direction.