**2.7.1 Multiplication by Scalar: **Let **a** be a vector quantity and ά be a scalar quantity. Then,

**b** = ά **a**

is a vector quantity having direction parallel to a and its magnitude is given by;

|**b|** = ά |**a|**

A unit vector **n **parallel to **a** is given by;

--

|a|

2.7.1.1 Product of vector and scalar in Cartesian components:<

**a** = **a _{x}i + a_{y}j + a_{z}k**

similarly,

**b **= **b _{x}i + b_{y}j + b_{z}k **

and we have,

**b** = ά **a**

therefore,

** b _{x}i + b_{y}j + b_{z}k = **ά (

hence,

**b _{x} = **ά

**2.7.2 Vector products:** Two other powerful methods that provide easy working with vectors are the dot (or inner) product and the cross product. The dot product produces a scalar and the cross product operates only on 3−dimensional vectors and produces another vector.

**2.7.2.1 Dot Products: **The **scalar product** of two vectors **A** and **B** is denoted by

**A** · **B**.

The scalar product is also called the **dot product.**

To define the scalar product **A** · **B** of two vectors **A** and **B**, draw the two vectors with their tails at the same point as shown in the figure below.

The angle between their directions is φ as shown; the angle φ always lies between 0° and 180°. (As usual, we use Greek letters for the angles.) The figure below shows the projection of vector **B** onto the direction of **A**;

This projection is the component of **B** parallel to **A** and is equal to B cos φ.

We define **A** · **B** to be the magnitude of **A** multiplied by the component of **B** parallel to **A**

**A** · **B** = AB cos φ = |**A**||**B**|cos φ

where φ ranges from 0° and 180°.

Alternatively, we can define **A** · **B** to be the magnitude of **B** multiplied by the component of **A** parallel to **B**,

which is the same as the previous equation above.

When φ is between 0° and 90°, the scalar product is positive.

When φ is between 90° and 180°, it is negative.

When φ = 90°, **A** · **B** = 0. The scalar product of two perpendicular vectors is always zero.

For any two vectors **A** and **B**,

AB cos φ = BA cos φ.

This means that

**A** · **B** = **B** · **A**.

The scalar product follows the commutative law of multiplication; the order of the two vectors does not matter.

The scalar product can also be expressed in terms of components as follows:

**A** · **B** = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}

The dot product is distributive:

and commutative:

The angles between i, j, and k vectors is π/2 radians (90 degrees) and cos π /2 = 0, we can derive a handy alternative definition: Let,

**u = ai + bj + ck
v = xi + yj + zk
**then,

=>u.v = (ai + bj + ck). xi + (ai + bj + ck).yj + (ai + bj + ck).zk

The angle between any nonzero vector and itself is 0, and cos 0 = 1, so i.i = 1 etc., Hence,

**u****.v = a x + b y + c z **

This means that for any vector, a,

**a****. a = a ^{2}**

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