Resolution of Vectors

We can now resolve a vector A in terms of component vectors that lie along unit vectors î and ĵ. Consider a vector A that lies in x - y plane as shown in figure. We draw lines from the head of A perpendicular to the coordinate axes as in figure and get vectors A₁ and A₂ such that A₁ + A₂ = A since A₁ is parallel to î and A₂ is parallel to ĵ. we have:

A₁ = A_x î, A₂ = A_y ĵ

Where A_x and A_y are real numbers.

Thus, A = A_x î + A_y ĵ

This is represented in figure. The quantities A_x and A_y  are called x - and y - components of the vector A. note that A_x is itself not a vector, but A_x î  is a vector, and so is A_y ĵ. Using simple trigonometry, we can express A_x and A_y in terms of the magnitude of A and the angle θ  it makes with the axis:

A_x = A cos θ

A_y = A sin θ

As is clear from A_y = A sin θ a component of a vector can be positive, negative or zero depending on the value of θ.

Now, we have two ways to specify a vector A in a plane. It can be specified by:

i) Its magnitude A and the direction θ it makes with the x -axis or

ii) Its components A_x and A_y

If A and θ are given, A_x and A_y can be obtained using A_y = A sin θ. If Ax and A_y are given, A and θ can be obtained as follows:

A²_x + A²_y = A² cos² θ + A² sin² θ

A²_x + A²_y = A²

Or 

A = √ (Aₓ² + A_y²)

And tanθ = A_y/Aₓ, θ = tan⁻¹ A_y/Aₓ

A_x = A cosα, A_y = A cos β, A_z = A cos ɣ

In general, we have

A = A_x î + A_y ĵ + A_z } } k̂

The magnitude of vector A is A = √ (Aₓ² + A_y² + A_z²)

A position vector r can be expressed as r = x î + y ĵ + z} } k̂.

Where x, y and z are the components of r along x, y, z- axes, respectively.} }