Vectors

A vector describes a movement from one point to another. It is a mathematical quantity having both Magnitude & Direction. The length of the segment of the directed line is called the magnitude of the vector and the angle at which the vector is inclined shows the direction of the vector.

Vectors

The beginning point of a vector is called as “Tail” and the end side (having arrow) is called as “Head.”

Examples & Representation-

Velocity, Acceleration, Force, Increase/Decrease in Temperature etc.

Vector Representation

A vector between two points A and B is given as \(\overrightarrow{AB}\), or vector a.

Understanding more about Vectors-

Breaking a vector into its x and y components is the most common way for solving vectors.

Component of Vectors

A vector “a” is inclined with horizontal having an angle equal to \(\theta\).

This given vector “a” can be broken down into two components i.e. ax and ay.

The component ax is called as “Horizontal component” whose value is \(a \cos \theta\).

The component ay is called as “Vertical component” whose value is \(a \sin \theta\).

Example- Given vector V, having magnitude of 10 units & inclined at \(60^{\circ}\). Break down the given vector into its two component.

Solution- \(\overrightarrow{v} \), having magnitude(V) = 10 units and \(\theta = 60^{\circ}\)

Horizontal component (Vx) = \(V \cos \theta\)

Vx = \(10\; \cos 60^{\circ}\)

Vx = \(10 \times 0.5\)

Vx = 5 units

Now, Vertical component(Vy) = \(V \sin \theta\)

Vy = \(10\; \sin 60^{\circ}\)

Vy = \(10 \times \frac{\sqrt{3}}{2}\)

Vy = \(10 \sqrt{3}\) units

Magnitude of a Vector-

The magnitude of a vector is shown by vertical lines on both the sides of the given vector.

\(\left | a \right |\)

Mathematically, the magnitude of a vector is calculated by the help of “Pythagoras Theorem,” i.e.

\(\left | a \right | = \sqrt{x^{2}+y^{2}}\),

Example- Find the magnitude of vector a (3,4).

Solution-

Given \( \overrightarrow{a} \)= (3,4)

\(\left | a \right |= \sqrt{x^{2}+y^{2}}\)

\(\left | a \right |= \sqrt{3^{2}+4^{2}}\)

\(\Rightarrow \left | a \right |= \sqrt{9 + 16} = \sqrt{25} \)

Therefore, \(\left | a \right |= 5\)

Operation on Vector

Vector operation such as Addition, Subtraction, Multiplication etc. can be done easily.

1. Addition of Vectors-

The two vectors a and b can be added giving the sum to be a + b. This requires joining them head to tail.

Addition of Vectors

We can translate the vector b till its tail meets the head of a. The line segment that is directed from the tail of vector a to the head of vector b is the vector “a + b”.

Characteristics of Vector Addition-

  • Commutative Law- the order of addition does not matter, i.e, a + b = b + a
  • Associative law- the sum of three vectors has nothing to do with which pair of the vectors is added at the beginning.

i.e. (a + b) + c = a + (b + c)

2. Subtraction of Vectors-

Before going to the operation it is necessary to know about reverse vector(-a).

Subtraction of Vectors

A reverse vector (-a) which is opposite of a has similar magnitude as a but pointed in opposite direction.

First, we find the reverse vector.

Then add them as the usual addition.

Such as if we wanna find vector b – a

Then, b – a = b + (-a)

3. Multiplication of Vectors-

  • Scalar Multiplication-

Multiplication of a vector by a scalar quantity is called “Scaling.”

In this type of multiplication, only the magnitude of a vector is changed not the direction.

  • Vector Multiplication-

It is of two types “Cross product” and “Dot product.”

Cross Product-

The cross product of two vectors results in a vector quantity. It is represented by a cross sign between two vectors.

  \(a \times b\)

Mathematical value of a cross product-

\(a \times b = \left | a \right | \left | b \right |\sin \theta \;\hat{n}\)

where, \(\left | a \right |\) is the magnitude of vector a.

\(\left | a \right |\) is the magnitude of vector b.

\(\theta\) is the angle between two vectors a & b.

and \(\hat{n}\) is a unit vector showing the direction of the multiplication of two vectors.

Dot product-

The dot product of two vectors always result in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot in between two vectors.

\(a.b\)

Mathematical value-

\(a . b = \left | a \right | \left | b \right |\cos \theta\)
Example- Find the scalar and vector multiplication of two vectors a and b given by \(3\hat{i}-1\hat{j}+2\hat{k}\) and \(1\hat{i}+-2\hat{j}+3\hat{k}\) respectively.

Solution-

Given vector a (3,-1,2) and vector b (1,-2,3)

Vector product ( or Cross product) = \(\vec{a} \times \vec{b} = \begin{vmatrix} i & j & k \\ 3 & -1 & 2 \\ 1 & -2 & 3 \end{vmatrix}\)

\(\vec{a} \times \vec{b} = \begin{vmatrix} -1 & 2 \\ -2 & 3 \end{vmatrix} \hat{i} – \begin{vmatrix} 3 & 2\\ 1 & 3 \end{vmatrix}\hat{j} + \begin{vmatrix} 3 & -1\\ 1 & -2 \end{vmatrix}\hat{k}\)

\(\vec{a} \times \vec{b} = 1 \hat{i} – (7)\hat{j} + (-5) \hat{k}\)

\(\vec{a} \times \vec{b} \) = \( 1 \hat{i} -7 \hat{j} – 5 \hat{k}\)

Now, \(\left | \vec{a} \times \vec{b} \right | = \sqrt{(1)^{2}+ (-7)^{2}+ (-5)^{2}}\)

\(\left | \vec{a} \times \vec{b} \right |\) = \(\sqrt{75} = 5\sqrt{3}\)

Now finding magnitudes of vector a and b

\(\left | \vec{a} \right | = \sqrt{(3)^{2}+(-1)^{2}+(2)^{2}}\)

\(\left | \vec{a} \right |\) = \(\sqrt{9+1+4} =\sqrt{14}\)

\(\left | \vec{b} \right | = \sqrt{(1)^{2}+(-2)^{2}+(3)^{2}}\)

\(\left | \vec{b} \right |\) = \(\sqrt{1+4+9} =\sqrt{14}\)

\(\sin \theta\) = \(\frac{\left | \vec{a} \times \vec{b} \right | }{\left | \vec{a} \right | \times \left | \vec{b} \right | }\)

\(\sin \theta = \frac{5\sqrt{3} }{\sqrt{14} \times \sqrt{14}}\)

\(\sin \theta = \frac{5\sqrt{3} }{14}\)

Or, \(\theta\) = \(\sin^{-1}\left ( \frac{5\sqrt{3} }{14} \right )\)

Thus the Vector product is equal to \(1 \hat{i} -7 \hat{j} – 5 \hat{k} \)

Scalar product (or Dot product) = \(\vec{a}.\vec{b}= \left | a \right |\left | b \right | \cos \theta\)

Where \(\theta\) is the angle between the vectors. But we don’t know the angle between the vectors thus another method of multiplication can be used.

\(\vec{a}.\vec{b}= (3\hat{i}-1\hat{j}+2\hat{k}).(1\hat{i}-2\hat{j}+ 3\hat{k})\)

\(\vec{a}.\vec{b}= 3(\hat{i}.\hat{i})+2(\hat{j}.\hat{j})+6(\hat{k}.\hat{k})\)

\(\vec{a}.\vec{b}= 3+2+6\)

\(\vec{a}.\vec{b}= 11\)

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Vector

Vector in physics can be defined as a quantity comprised of both direction and magnitude. Apart from direction and magnitude vector not have any position, in simple it is not altered unless and until it is not displaced parallel to itself. A vector is represented by a symbol arrow whose length will be proportional to the quantity’s magnitude and lies in the same direction of the quantity.

Some of the vector quantities include force, displacement, acceleration and distance. Scaled vector diagrams are used to represent vector quantities. The magnitude of the vector is usually represented by the length of the arrow in a scalar vector diagram.

Representation of Vectors: Take velocity vector for an instant, if we want to represent a velocity vector of magnitude five units and along the direction of a positive x axis. This can be represented by drawing a line parallel to velocity and putting an arrow showing the direction of velocity.

vector

The vectors are denoted by putting an arrow over the symbols representing them. Thus, we write \(\vec{AB}\), \(\vec{BC}\) etc. Sometimes a vector is represented by a single letter such as:

Force vector: \(\vec{F}\)

Velocity vector: \(\vec{v}\)

Acceleration vector: \(\vec{a}\)

Linear momentum: \(\vec{p}\)

Equality of Vectors

Two vectors are said to be equal if their magnitudes and directions are same. Here we are talking about two values of the same physical quantity, i.e. we can not talk about equality of two vectors if they don’t represent the same physical quantity. For instance, one can’t say that velocity vector of 5 m/s in the positive x-axis and Force vector of 5 N also in positive x-axis are equal.


Practise This Question

Can a vector have zero component along a line and still have non-zero magnitude?