Vectors : Vectors are those physical quantities which have magnitude, fix direction and follows vector laws of addition.
For example; velocity, acceleration, momentum, force etc.
(a) Polar vectors: These are those vectors which have a starting point or a point of application.
For example; displacement, velocity, acceleration, force etc. are polar vectors.
(b) Axial vectors: These are those vectors which represent rotational effect and act along the axis of rotation in accordance with right hand screw rule. Ex of axial vectors are angular displacement, angular velocity, angular acceleration, angular momentum, torque etc.
What is the Importance Of Vectors ?
(1) Many laws of physics can be expressed in compact form by the use of vectors.
(2) The derivations involving many laws of physics can be greatly simplified by the use of vectors.
(3) The laws of physics when expressed in vector form remain invariant for translation and rotation of the coordinate system.
A Few Definitions In Vector Algebra
Parallel vectors : Those vectors which have equal or unequal magnitude but have same direction are called parallel vectors. Angle between anti parallel vectors is always 0°.
Anti-parallel vector : Those vectors which have equal or unequal magnitude but opposite direction are called anti parallel vector. Angle between parallel vectors is always 180°.
Modulus of vector : The magnitude of a vector is called modulus of that vector.
The modulus of a vector $\vec{A}$ is represented by $ |\vec{A} | $ or A.
Equal vectors : Two vectors are said to be equal if they have equal magnitude and same direction.
Negative/opposite vectors. A negative vector of a given vector is a vector of same magnitude but acting in a direction opposite to that of the given vector.
Unit vector : A unit vector of the given vector is a vector of unit magnitude and has the same direction as that of the given vector.
A unit vector of $\vec{A}$ is written as $\hat{A}$ and is read as ‘A cap’ or ‘A hat’. Since, magnitude of $\vec{A}$ is A,
Hence, $\vec{A} = A \hat{A}$
or $\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{vector}{modulus \; of \; vector}$
Thus, a unit vector in a given direction is also defined as a vector in that direction divided by the magnitude of the given vector. It is unitless and dimensionless vector and represents direction only.
In Cartesian coordinates, $\hat{i} \; , \hat{j} \; , \hat{k}$ are the unit vectors along X-axis, Y-axis and Z-axis respectively.
Co-initial vectors : the vectors are said to be co-initial, if their initial point is common.
Collinear vectors: These are those vector which are having equal or unequal magnitudes and are acting along the parallel straight lines.
Coplanar vectors: These are those vector which are acting in the same plane.
Concurrent vector : Vectors which passes through the same point are called concurrent vectors.
Localized vectors: It is that vector whose initial point is fixed. It is also called fixed vectors.
Non-localised vector: It is that vector whose initial point is not fixed. It is also called a free vector.
Resultant Vector (Addition of two vectors)
The resultant vector of two or more vectors is defined as that single vector which produces the same effect as is produced by individual vectors together.
It is to be noted that the nature of the resultant vector is the same as that of the given vectors.
Vectors Addition By Geometrical Method
General rule for addition of vectors. It states that the vectors to be added are arranged in such a way so that the head of first vectors coincides with the tail of second vector, whose head coincides with the tail of third vector and so on, then the single vector drawn from the tail of the first vector to the head of the last vector represents their resultant vector.
(a) When the two vectors are acting in the same direction.
$\large \vec{R} = \vec{A} + \vec{B} $
(b) When two vectors are acting in opposite direction.
$\large \vec{R} = \vec{A} – \vec{B} $
(c)Triangle law of vectors : Triangle law of vectors states that is two vectors acting on a particle at the same time are represented in magnitude and direction by the two sides of triangle taken in one order, their resultant vector is represented in magnitude and directed by the third side of the triangle taken in opposite order.
Parallelogram law of vectors : Parallelogram law of vectors states that if two vectors acting on a particle at the same time are represented in magnitude and direction by the two adjacent side of a parallelogram drawn from a point, their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
(c) When number of vectors act in different directions. Thus, polygon law of vectors
states that if any number of vectors, acting on a particle a the same time are represented in magnitude and direction by various sides of an open polygon taken in the same order, their resultant is represented in magnitude and direction by the closing side of the polygon taken in opposite order.
Note:
(1) If a number of vectors are represented by the various sides of a closed triangle ,rectangle or in general a polygon taken in one order, their resultant is zero.
(2) Angle between two vectors is found by joining their heads or tail together .
Parallelogram law of vectors :
Parallelogram law of vectors states that if a point (particle) is acted upon by two vectors which can be represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is completely represented in magnitude and direction by the diagonal of the parallelogram passing through the same point.
Suppose two vectors $\vec{P}$ and $\vec{Q}$ acting on a particle are represented by the sides OA and OB, inclined to each other at angle θ, then on completing parallelogram OACB, diagonal OC gives in magnitude and direction of the resultant of the vectors $\vec{P}$ and $\vec{Q}$ . From C draw CD perpendicular to OA produced. Then by Pythagoras theorem for right angled triangle OCD, we have
$\displaystyle OC^2 = OD^2 + CD^2 $
$\displaystyle OC^2 = (OA + AD)^2 + CD^2 $
$\displaystyle OC^2 = (OA^2 + AD^2 + 2 (OA)(AD)) + CD^2 $
$\displaystyle OC^2 = OA^2 + (AD^2 + CD^2 )+ 2 (OA)(AD) $
$\displaystyle OC^2 = OA^2 + AC^2+ 2 (OA)(AD) $
Since , $ \displaystyle cos\theta = \frac{AD}{AC} $
$\displaystyle OC^2 = OA^2 + AC^2+ 2 (OA)(AC cos\theta) $
$\displaystyle R^2 = P^2 + Q^2+ 2 P Q cos\theta $
$\displaystyle R = \sqrt{P^2 + Q^2+ 2 P Q cos\theta } $
$\displaystyle tan\alpha = \frac{Q sin\theta}{P + Q cos\theta}$
( where α is the angle between first vector P and the resultant )
Special Cases :
(i) If θ = 0°
$\displaystyle R = \sqrt{P^2 + Q^2+ 2 P Q cos0^o } = P + Q$
(ii) If θ = 180 °
$\displaystyle R = \sqrt{P^2 + Q^2+ 2 P Q cos0180^o } = P – Q$
(iii) If θ = 90 °
$\displaystyle R = \sqrt{P^2 + Q^2+ 2 P Q cos90^o } = \sqrt{P^2 + Q^2 } $
Maximum and minimum magnitude of the resultant of two vectors $\vec{P}$ and $\vec{Q}$ are given by putting θ = 0° and θ = 180° in expression
$\displaystyle R = \sqrt{P^2 + Q^2 + 2 P Q cos\theta} $
$\displaystyle R_{max} = \sqrt{P^2 + Q^2+ 2 P Q cos0^o }$
$\displaystyle R_{max} = P + Q $
$\displaystyle R_{min} = \sqrt{P^2 + Q^2+ 2 P Q cos180^o }$
$\displaystyle R_{min} = P – Q $
Solved Example : If two vectors of equal magnitudes have their resultant equal to the magnitude of either of them, then find the angle between them ?
Solution: $\displaystyle R^2 = P^2 + Q^2+ 2 P Q cos\theta $
Given , P = Q = R
$\displaystyle R^2 = R^2 + R^2+ 2 R^2 cos\theta $
$\displaystyle cos\theta = -\frac{1}{2}$
θ = 120°
Triangle law of Vectors
If a particle is simultaneously acted upon by two vectors which can be represented by the two sides of a triangle taken in order then the closing side of the triangle taken in opposite order gives the resultant both in magnitude as well as in direction, i.e.,
$\displaystyle \vec{A} + \vec{B} = \vec{C} $
Resolution Of Vectors :
(a) Rectangular resolution of a vector in a plane :
$\displaystyle \vec{A} = A_x \hat{i} + A_y \hat{j} $
If this vector makes an angle θ with X-axis then it can be proved that Ax = A Cosθ and Ay = A Sin θ
And , $\displaystyle A = \sqrt{A_x^2 + A_y^2 }$
(b) Rectangular resolution of a vector in space
Let , $\displaystyle \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} $
If this vector makes an angle α with X-axis , β with the Y axis and γ with the Z axis then :
Ax = A Cosα , Ay = A Cosβ , Az = A Cosγ
And , $\displaystyle A = \sqrt{A_x^2 + A_y^2 + A_z^2 }$
Cosα , Cosβ , Cosγ are known as direction cosine also it can be proved that
$\displaystyle Cos^2 \alpha + Cos^2 \beta + Cos^2 \gamma = 1 $
$\displaystyle L.H.S = Cos^2 \alpha + Cos^2 \beta + Cos^2 \gamma $
$\displaystyle = (\frac{A_x}{A})^2 + (\frac{A_y}{A})^2 + (\frac{A_z}{A})^2 $
$\displaystyle = \frac{A_x^2 + A_y^2 + A_z^2 }{A^2} $
$\displaystyle = \frac{A^2}{A^2} = 1 $
Note:
(i) Any vector can be resolved in infinite components
(ii) Any vector in a plane ,can be resolved in maximum two rectangular components.
(iii) Any vector in space can be resolved in maximum three rectangular components.