Question 3: What are rectangular components of a vector? How rectangular components are used to represent a vector?
Resolution of a vector in components
Replacing a vector by two or more than two vectors is called the resolution of vectors. The vectors so obtained are called the components of the original vector. Combined effect of the component vectors is the same as the original vector.
If the components of the vector are mutually perpendicular then they are called the rectangular components of the original vector.
REPRESENTATION OF A VECTOR BY RECTANGULAR COMPONENTS
A vector is the combined effect of its components; therefore, the vector can be represented in terms of its rectangular components.
Let’s consider a vector A represented by line OP in the Cartesian plane as shown in the figure. In order to resolve this vector into its rectangular components, draw a perpendicular from the tip of the vector on x-axis. Let it meets the x-axis on point Q. OQ is along the x-axis and called the x-component of vector A, represented by Ax.
Then draw perpendicular from the tip of the vector A on y-axis and let it meets the y-axis at point S. OS is along y-axis and called the y-component of vector A. It is represented by Ay.
Now by the definition of the (rectangular) components,
Here î and ĵ are the unit vectors along x- and y-axis and the equation represents A in terms of its rectangular components.
Rectangular components of a vector can also be represented in terms of trigonometric ratios and therefore, the vector may also be represented as sum of the trigonometric ratios.
Consider the ΔOPQ as shown in the figure.
Sinθ = PQ/OP ⇒ PQ = OPsinθ … (i)
Cosθ = OQ/OP ⇒OQ = OPcosθ … (ii)
Compare with the above figure (first one above) , we see
PQ = Ay = Asinθ … (iii)
OQ = Ax = Acosθ … (iv)
Put these values in equation (1),
This equation shows the vector represented in terms of its rectangular components.
Magnitude and direction of the vector can be easily calculated from its rectangular components.
Magnitude calculation from the rectangular components
By Pythagoras theorem of right angled triangle,
(hypot)2 = (base)2 + (perp)2, therefore, from the above fig (1),
Equation (A) gives the magnitude of the vector in terms of its rectangular components.
Direction calculation from the rectangular components
The direction of the vector can be calculated from the formula of a right angled triangle.
tanθ = perpendicular/base
Perpendicular and base forms the y- and x-components of the vector, respectively. Therefore,
Equation (B) gives the angle the vector makes with the x-axis in terms of its components.