## Resolution of a Vector in a Plane:

Let **a **and **b **be any two nonzero vectors in a plane with different directions and let **A **be another vector in the same plane. **A **can be expressed as a sum of two vectors – one obtained by multiplying **a **by a real number and the other obtained by multiplying **b **by another real number. To see this, let O and P be the tail and head of the vector **A**. Then, through O, draw a straight line parallel to **a**, and through P, a straight line parallel to **b**. Let them intersect at Q. Then, we have

**A **= **OP **= **O****Q **+ **Q****P**

But since **O****Q **is parallel to **a**, and **Q****P **is parallel to **b**, we can write:

**OQ **= λ **a**, and **QP** = μ**b**

where λ and *µ *are real numbers.

Therefore, **A **= λ **a **+ *µ ***b**

### Resolution of a Vector into Rectangular Components:

If any vector A subtends an angle θ with x-axis, then its

Horizontal component Ax = A cos θ

Vertical component Ay = A sin θ

**Magnitude of vector**

### Direction Cosines of a Vector:

If any vector A subtend angles α, β and γ with x – axis, y – axis and z – axis respectively and its components along these axes are A_{x}, A_{y} and A_{z} then

## Unit Vector:

A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only. Unit vectors along the *x*-, *y *and *z*-axes of a rectangular coordinate system are denoted by ** **î, ĵ** **and k respectively,

Since these are unit vectors, we have

These unit vectors are perpendicular to each other.

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