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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Practise This Question

What direction must you swim at to cross the river in minimum time given the crocodiles have left and your maximum velocity is 5Kmhr and the velocity of river is 3Kmhr in the west direction.