Question

What do you understand by rectangular resolution of a vector? Resolve it into its two rectangular components.

Answer 1

It is defined as the factors of splitting given vector in two or three component vectors at right angle to each other. The component vectors are called rectangular component of the given vector. Let \(\vec R\) be the given vector acting in X-Y plane at an angle θ with X-axis. Let \(\vec a\) = \(\vec R\) from point C, draw CA and CB on X and Y axes respectively. If \(\vec P\) and \(\vec Q\) be rectangular components of \(\vec R\) along X and Y axes respectively, then

\(\vec {OA}\) = \(\vec P\) or OA = P 

and \(\vec {OB}\) = \(\vec Q\)or OB = Q 

Now in right angled ∆OAC, 

sin θ = \(\frac{AC}{OC}\) = \(\frac {Q}{P}\) 

or Q = R sin θ …(i)

and cos θ = \(\frac{AC}{OC}\) = \(\frac PR\)

or P = R cosθ …(ii)

 Also OC² = OA² + AC² 

or R ² = P² + Q²

 or R = \(\sqrt {P^2 + Q^2}\)

and tan θ =\(\frac {AC}{OA}\) =\(\frac {Q}{P}\) 

Also according to ∆ law of vector addition, 

\(\vec R\) = \(\vec P\) + \(\vec Q\) 

= \(P \hat i + Q \hat j\)

or \(\vec R\) = (R cos θ)\(\hat j\)+ (R cos θ)\(\vec i\)

 Thus if \(\vec {A_x} \) and \(\vec {A_y}\)  be the two rectangular components of \(\vec A\) along X and Y axes respectively,

then \(\vec A\) = \(\vec {A_x} \)\(\hat i\)+ \(\vec {A_y}\)\(\hat j\)