__Vector in space__

__Vector__

The vectors are defined as an object containing both magnitude and direction. Vector describes the movement of an object from one point to another. **Vector math** can be geometrically picturized by the directed line segment. The length of the segment of the directed line is called the magnitude of a vector and the angle at which the vector is inclined shows the direction of the vector. The beginning point of a vector is called “Tail” and the end side (having arrow) is called “Head.”

**Subtracting**

We can also subtract one vector from another:

- first, we reverse the direction of the vector we want to subtract,
- then add them as usual:

## Adding Vectors

We can then add vectors by **adding the x parts** and **adding the y parts**:

**Example: add the vectors **a** = (8,13) and **b** = (26,7)**

**c** = **a** + **b**

**c** = (8,13) + (26,7) = (8+26,13+7) = (34,20)

## Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

**Example: subtract **k** = (4,5) from **v** = (12,2)**

**a** = **v** + −**k**

**a** = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)

## Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

|**a**|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||**a**||

We use Pythagoras’ theorem to calculate it:

|**a**| = √( x^{2} + y^{2} )

**Example: what is the magnitude of the vector **b** = (6,8) ?**

|**b**| = √( 6^{2} + 8^{2 }) = √( 36+64^{ }) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

## Multiplying a Vector by a Scalar

When we multiply a vector by a scalar it is called “scaling” a vector, because we change how big or small the vector is.

**Example: multiply the vector **m** = (7,3) by the scalar 3**

a = 3m = (3×7,3×3) = (21,9) |

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called “scalars”, because they “scale” the vector up or down.)

**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.

- S(a+b) = Sa + Sb
- (S+T)a = Sa + Ta
- a.1 = a
- a.0 = 0
- a.(-1) = -a

**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 × b**

The mathematical value of a cross product-

where,

| a | is the magnitude of vector a.

| b | is the magnitude of vector b.

θ is the angle between two vectors a & b.

and n^ is a unit vector showing the direction of the multiplication of two vectors.

**Dot product**

The dot product of two vectors always results in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot in between two vectors.

a.b

The mathematical value of the dot product is given as

a . b = | a | | b | cos θ |

**IF A= A _{x} a_{x}+ A _{y}a_{y} + A_{z}a_{z}**

__Unit Vector__

A unit vector is defined as a vector in any specified direction whose magnitude

is unity i.e. 1. A unit vector only specifies the direction of a given vector.

In three dimensional coordinate system unit vectors having the direction of the positive X-axis, Y-axis and Z-axis are used as unit vectors. These unit vectors are mutually perpendicular to each other.

__Null Vector__

A null vector is a vector having a magnitude equal to zero. It is represented by. A null vector has no direction or it may have any direction. Generally, a null vector is either equal to resultant of two equal vectors acting in opposite directions or multiple vectors in different directions. |