Vectors

Vector Definition

Mathematical representation of physical quantities for which both magnitude and direction can be determined is called vector. Vector of any physical quantity is represented as a straight line with an arrow head. In vector definition, the length of the straight line denotes the magnitude of the vector and the arrow head gives its direction. Any two vectors can be regarded as identical vectors if they have equal magnitude and direction. The best example for a vector is the force applied on an object because both strength and direction of applied force affects its action on the object. Rotating or moving a vector around itself will never change its magnitude. Image will be uploaded soon.

 

Vector Math

Vector Math finds a wide range of applications in various domains of Algebra, Geometry and Physics. As discussed above, a vector is represented as a straight line with an arrow head. The end points of a vector are generally labeled with letters of English Alphabet in uppercase. Vectors are symbolically represented as the end points with an arrow head or a lower case letter with an arrow head. Image will be uploaded soon.

 

In the above vector, the region enclosed by a flower bracket indicates the magnitude of the vector and the arrow head indicates vector direction. This vector is symbolically represented as \[\overline{AB}\] or \[\overrightarrow{a}\]. The magnitude of this vector is given as |AB| or |a|. It represents the vector length and is generally calculated with the help of Pythagorean theorem. The basic Mathematical operations like addition, subtraction and multiplication can be performed on vectors. However, division of two vectors is not possible.


Vocabulary of Vectors:

Most important terms associated to vectors are:

  1. Zero Vector :  A vector whose magnitude is zero. 

  2. Unit Vector :  A vector with magnitude of one unit.

  3. Position Vector : A vector that denotes the position of a point with respect to its origin.

  4. Co Initial Vector : Two or more vectors with the same starting point.

  5. Like and Unlike Vectors : Vectors with the same direction are called like vectors and those with different directions are called unlike vectors.

  6. Coplanar Vectors : Vectors in the same plane.

  7. Collinear Vector : Vectors lying on the same straight line.

  8. Equal Vectors : Two or more vectors with the same magnitude and direction.

  9. Displacement Vector :  A vector indicating displacement of an object from one point to another.

  10. Negative of a Vector : Negative of any vector is another vector with the same magnitude but opposite direction.

 

Mathematical Operations on Vector

1. Vector Addition

Vector Addition is performed on any two vectors using triangle law of vector addition. According to this law, the two vectors to be added are represented by two sides of a triangle with the same magnitude and direction. The third side gives the magnitude and direction of the resultant addition vector. Image will be uploaded soon.


2. Vector Subtraction

Considering two vectors a and b. If vector ‘a’ is to be subtracted from vector ‘b’, negative of vector ‘a’ has to be found and it should be added to vector ‘b’  using triangle law.


b - a = b +(-a)


3. Vector Multiplication

Multiplication of any two vectors is performed by finding their ‘cross product’ or ‘dot product’.


  • Cross Product of Two Vectors a and b is Mathematically Calculated as:

a b =|a||b| sin θ n̂

where, |a| is the magnitude of vector ‘a’

|b| is the magnitude of vector ‘b’

θ is the angle of separation of two vectors ‘a’ and ‘b’

n̂ is the unit vector representing the direction of multiplication of vectors


  • Vector Definition of Dot Product of Two Vectors a and b is Mathematically Given as:

a . b =|a||b| cos θ

where,  |a| is the magnitude of vector ‘a’

|b| is the magnitude of vector ‘b’

θ is the angle of separation of two vectors ‘a’ and ‘b’

 

  • Cross product of two vectors is a vector quantity. It has both magnitude and direction whereas, the dot product of two vectors has only magnitude and no direction. So, it is a scalar quantity. 

 

Vector Mathematics Examples:

1. Find the resultant addition vector of vector a= (8,13) and vector b=(12, 15).

Solution: The addition vector of ‘a’ and ‘b’ obtained as 

c = a+b

c = (8, 13) +(12, 15)

c = (8+12)+(13+15)

c = (20, 27)

 

2. In one of the vector questions, k = (3, 4) and m = (7, 9). Subtract vector ‘k’ from vector ‘m’.

Solution: To subtract vector ‘k’ from vector ‘m’, the negative vector of ‘k’ should be found.

Negative vector of ‘k’ = - k

= - (3, 4)

= ( -3, -4)

The subtraction of vector ‘k’ from vector ‘m’ is given as:

m - k = m + (-k)

= (7, 9) + (-3, -4)

= (7 - 3), (9 - 4)

= (4, 5)


3. Determine the magnitude of vector c = (5, 12)

Solution: Magnitude of vector ‘c’ is calculated as,

|c| = \[\sqrt{x^{2} + y^{2}}\]

|c| = \[\sqrt{5^{2} + 12^{2}}\]

|c| = \[\sqrt{25 + 144}\]

|c| = \[\sqrt{169}\]

|c| = 13 units


4. In one of the vector Mathematics examples, if |a| = 5 units and |b| = 10 units, find the dot product if the angle of separation between vector ‘a’ and ‘b’ is 60o.

Solution: Dot product of two vectors can be calculated as:

\[a \cdot b = |a||b| cos \theta\]

\[a \cdot b = 5\times 10\times cos 60^{o}\]

\[a \cdot b = 50\times \frac{1}{2}\]

\[a \cdot b = 25 \text{units}\]


5. Compute cross product of 2 vectors ‘k’ and ‘l’ whose magnitudes are 7 units and 9 units respectively if the angle between the two vectors is 90o.

Solution: Dot product of two vectors can be calculated as:

\[a \cdot b = |a||b| cos \theta\]

\[a \cdot b = 7\times 9\times cos 90^{o}\]

\[a \cdot b = 63 \times 0\]

\[a \cdot b = 0 \text{units}\]


Fun facts:

  • Any geometric object which has both magnitude and direction is called an Euclidean Vector.

  • Matrices can also be used with the help of vector definition. Any matrix with a single row or a single column is termed as row vector or column vector respectively.

FAQ (Frequently Asked Questions)

1. Distinguish Between Scalars and Vectors.

Ans. 

Scalars

Vectors

Scalars are physical quantities with only magnitude and no direction.

Vectors are the physical quantities which has both magnitude and direction

Among any two scalars, all basic Mathematical operations can be performed including addition, subtraction, multiplication and division.

Addition, subtraction and  multiplication can be performed on vectors. However, vector division is not practical in vector questions.

Scalars do not have negatives.

While finding vector Maths solutions, negative of a vector is taken as another vector of the same magnitude and opposite direction.

Eg: Length, mass, speed

Eg: Force, weight, velocity, acceleration are a few vector mathematics examples.

2. What are the Components of a Vector?

Vectors can be resolved into two or more smaller components by a process called resolution of vectors. Any vector can be resolved in a vector space into two components namely horizontal component and vertical component. The horizontal component is a product of the magnitude of the vector and the cosine of the horizontal angle. Vertical component of a vector is the product of the magnitude of the vector and the sine of its horizontal angle.

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The above figure depicts an example of vector resolution in which vector ‘a’ is resolved into two components ax and ay. The vector ax is the horizontal component of vector ‘a’ given by a Cosand ay is the vertical component given by a Sin. Vector resolution is generally used to simplify the calculations of vector operations.