Question

The rectangular components of a vector lying in XY plane are 1 and p+1. If the coordinate system is turned by $${30}^{\circ }$$, they are p and 4 respectively. The value of p is
A) 2
B) 4
C) 3.5
D) 7

Answer 1

Hint:
First of all, we’ll let the vector composition after turning by $${30}^{\circ }$$. We have given value of component after rotation. we’ll use then to get the final answer.

Complete step by step solution:
Given that components of vector lying in XY plane are 1 and p+1.
Let the components after rotation be x’ and y’ respectively.
x’= $$x\cos \theta +y\sin \theta$$
y’= $$y\cos \theta -x\sin \theta$$
but it is already given that after rotation of the system by $${30}^{\circ }$$, the coordinates become p and 4.
So,
p= $$x\cos \theta +y\sin \theta$$
But $$\theta$$=$${30}^{\circ }$$ , x coordinate =1 and y coordinate = p+1 is given
$$\begin{gathered} \Rightarrow p=1\cos {30}^{\circ }+\left(p+1\right)\sin {30}^{\circ } \\ \Rightarrow p={\displaystyle \frac{\sqrt{3}}{2}}+{\displaystyle \frac{\left(p+1\right)}{2}} \end{gathered}$$
Now ,
4=$$y\cos \theta -x\sin \theta$$
$$\begin{gathered} \Rightarrow 4=\left(p+1\right)\cos {30}^{\circ }-1\sin {30}^{\circ } \\ \Rightarrow 4={\displaystyle \frac{\left(p+1\right)\sqrt{3}}{2}}-{\displaystyle \frac{1}{2}} \end{gathered}$$
Solving this equation ,
$$\begin{gathered} \Rightarrow 4\times 2=\left(p+1\right)\sqrt{3}-1 \\ \Rightarrow 8=\left(p+1\right)\sqrt{3}-1 \\ \Rightarrow 8+1=\left(p+1\right)\sqrt{3} \\ \Rightarrow 9=\left(p+1\right)\sqrt{3} \\ \Rightarrow {\displaystyle \frac{9}{\sqrt{3}}}=\left(p+1\right) \\ \Rightarrow {\displaystyle \frac{3\times \sqrt{3}\times \sqrt{3}}{\sqrt{3}}}=\left(p+1\right) \\ \Rightarrow 3\sqrt{3}=p+1 \\ \Rightarrow p=3\sqrt{3}-1 \\ \Rightarrow p=4 \end{gathered}$$
Value of p=4.

Hence option B is correct.

Note:
1) Rectangular components are obtained from a vector itself, one of the x-axis and another on the y-axis.
2) If A is a vector then its x component is Ax and its y component is Ay.
3) Some angle is also resolved along with these vectors.