Laws of Vector

Multiplying a vector with a positive number n gives a vector (= n) whose magnitude is changed by the factor n, but the direction is same as that .

Also, Multiplying a vector with a negative number n gives a vector (= – n) whose direction is opposite to the direction of and its magnitude is – n times .

An example is given:

Diagram (a): Multiplication by a positive number

Diagram (b): Multiplication by a negative number

Figure:6.a

Vector addition and subtraction – graphical method:

Let OA = p be a vector. From the terminal point of p, another vector AB = q is drawn. Then, the vector OB from the initial point of p to the terminal point B of b is called the sum of vectors p and q and is denoted by s = p + q. This is called the triangle law of addition of vectors.

Figure:6.b

From the above diagram, we get,

s = p + q

Or,

Points to remember:

1. Vector addition is commutative i.e. p + q = q + p
2. Vector addition is associative i.e. p + q + s = (p + q) + s = p + (q + s) = (p + s) + q
3. p + 0 = p
4. p + (- p) = 0

Now, the above diagram can be expressed as below:

Figure:6.c

The direction of vector b is reversed and then the vector addition process is followed.

Then, it can be written as,

s = p – q = p + (- q)

Vector addition – analytical method:

Parallelogram law: If two vectors P and Q are represented by two adjacent sides of a parallelogram both pointing outwards, then the diagonal of the parallelogram represents the resultant of P and Q.

Figure:6.d

From the diagram, we get,

R = P + Q

BC is normal to OC.

From the geometry of the diagram,

$OB^{2}=OC^{2}+CB^{2}$ ……………………….. (1)

Now, AC = $Qcos\theta $

So, OC = P + $Qcos\theta $

And BC = $Qsin\theta $

From equation (1),

$R^{2}=(P + Qcos\theta )^{2}+$$(Qsin\theta )^{2}$

Or, $R=\sqrt{P^{2}+Q^{2}+2PQcos\theta }$……………………….. (2)

The direction of the resultant vector R from vector A is given by,

$\tan \varphi =BC/OC$

Or, $\tan \varphi =\frac{Qsin\theta }{P + Qcos\theta }$

Or, $\varphi =\tan ^{-1}(\frac{Qsin\theta }{P + Qcos\theta })$ ……………….. (3)

Equation (2) and (3) give the magnitude and direction of the resultant vector respectively.

Points to remember:

1. As $R=\sqrt{P^{2}+Q^{2}+2PQcos\theta }$, R will be maximum when $\cos \theta $= 1 i.e. $\theta =0$. This signifies that vectors are parallel. So, $R_{\max }=P+Q$.
2. R will be minimum when $\cos \theta $= – 1 i.e. $\theta =180$. This signifies that vectors are antiparallel. So, $R_{\min }=P-Q$.

Zero vector:

Zero vector is a vector whose initial and terminal points are same. The magnitude of the zero vector is always zero. This is also called null vector. The zero vector is denoted by .

When a zero vector is added or subtracted from a vector, then the result is the vector itself in each case.

Resolution of vectors:

If any two nonzero vectors are in a plane with different directions and be another vector in the same plane, then can be expressed as a sum of two vectors – one multiplying by a real number and the other by another real number.

, where $\alpha $and $\beta $ and are real numbers

So, vector has been resolved into two component vectors i.e. and .

Resolution in two dimensions:

Consider a vector vector lies in XY plane such that, 

Now,
, where are unit vectors along the X and Y – axis

Figure:6.e

So,

Or,

The magnitude $\vert a\vert =\sqrt{a_{x}^{2}+a_{y}^{2}}$

And direction, $\theta =tan^{-1}(a_{y}/a_{x})$

Multiplication of vectors:

Dot product:

The scalar product or dot product of any two vectors is written as .

The dot product is equal to the product of their magnitudes with cosine of angle between them.

Scalar product will be maximum when $\cos \theta $= 1, so, vectors are parallel.

Then,

Component of along c = $b cos\theta $

Or,

= 

Similarly, the component of along is 

 

Cross product:

The vector product or cross product of any two vectors is denoted by  .

It is defined as

Where $\theta $ is the angle between and ${n}$ is the unit whose direction is given by the right-hand thumb rule.

Figure:6.f

Stretch your right palm and pace perpendicular to the plane of in such a way that the finger is along the vector and towards  when fingers are closed. The direction of thumb gives the direction of ${n}$.

The vector product of two vectors is always a vector perpendicular to the plane containing the two vectors.

The vector product will be maximum if $\sin \theta $=1 or $\theta =90^{0}.$