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 = Ax î, A2= Ay ĵ
Where Ax and Ay are real numbers.
Thus, A = Ax î + Ay ĵ
This is represented in figure. The quantities Ax and Ay are called x – and y – components of the vector A. note that Ax is itself not a vector, but Ax î is a vector, and so is Ay ĵ. Using simple trigonometry, we can express Ax and Ay in terms of the magnitude of A and the angle θ it makes with the axis:
Ax = A cos θ
Ay = A sin θ
As is clear from Ay = 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 Ax and Ay
If A and θ are given, Ax and Ay can be obtained using Ay = A sin θ. If Ax and Ay are given, A and θ can be obtained as follows:
A2x + A2y = A2 cos2 θ + A2 sin2 θ
A2x + A2y = A2
Or A = √ (Aₓ² + Ay²)
And tanθ = Ay/Aₓ, θ = tan⁻¹ Ay/AₓAx = A cosα, Ay = A cos β, Az = A cos ɣ
In general, we have
A = Ax î + Ay ĵ + Az k̂
The magnitude of vector A is A = √ (Aₓ² + Ay² + Az²)
A position vector r can be expressed as r = x î + y ĵ + zk̂.
Where x, y and z are the components of r along x, y, z- axes, respectively.