Multiplying a vector with a positive number n gives a vector

Also, Multiplying a vector

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:*

- Vector addition is commutative i.e.
**p + q = q + p** - Vector addition is associative i.e.
**p + q + s = (p + q) + s = p + (q + s) = (p + s) + q** **p + 0 = p****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:**

- 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, $R_{\max }=P+Q$.
- 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

So, vector

*Resolution in two dimensions:*

Consider a vector

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

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

Or,

Similarly, the component of

## Cross product:

The vector product or cross product of any two vectors

It is defined as

Where $\theta $ is the angle between

*Figure:6.f*

Stretch your right palm and pace perpendicular to the plane of

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 $\theta =90^{0}.$