Question

If O=(0,0,0) , OP=5 and the d.rs for OP is 1,2,2 then the value of \[{{P}_{x}}+{{P}_{y}}+{{P}_{z}}=\]?
a) 25
b) \[\dfrac{25}{9}\]
c) \[\dfrac{25}{3}\]
d) \[\left( \dfrac{5}{3},\dfrac{10}{3},\dfrac{10}{3} \right)\]

Answer 1

Hint: To solve the type of question that has been mentioned we will first take the distance between O and P and then we will consider the value of d.rs in the form of l, m and n and then we will substitute it in the formula where P = (lr, mr, nr) and then with help of lr, mr and nr we will find the sum of the value required.

Complete step by step solution:
First we will take the value of OP that is the distance between point O and point P which is mentioned in the question as 5.
Then we will take the value of d.rs of OP which is given as l, m, and n. The mentioned values of l, m, and n in this question are as follows: l =1, m=2, n=2. Now that we gathered the information about the values of l, m, n we can proceed in finding the x, y and z coordinates of point P which we can calculate through the formula that can be applied when we know the values of l, m and n and also the distance between the two points O and P. the formula that we will be applying is P=(lr, mr, nr), on substituting the values in this formula we will get the x, y and z coordinates of point P which is what is required. So the values of \[{{P}_{x}}\], \[{{P}_{y}}\] and \[{{P}_{z}}\] are
\[\begin{align}
  & {{P}_{x}}=lr \\
 & \Rightarrow {{P}_{x}}=1\times 5 \\
 & \Rightarrow {{P}_{x}}=5 \\
\end{align}\]
\[\begin{align}
  & {{P}_{y}}=mr \\
 & \Rightarrow {{P}_{y}}=2\times 5 \\
 & \Rightarrow {{P}_{y}}=10 \\
\end{align}\]
 \[\begin{align}
  & {{P}_{z}}=nr \\
 & \Rightarrow {{P}_{z}}=2\times 5 \\
 & \Rightarrow {{P}_{z}}=10 \\
\end{align}\]
Now that we know the values of \[{{P}_{x}}\], \[{{P}_{y}}\] and \[{{P}_{z}}\] we can find the value of \[{{P}_{x}}+{{P}_{y}}+{{P}_{z}}\] which will result in as:
 \[\begin{align}
  & \Rightarrow {{P}_{x}}+{{P}_{y}}+{{P}_{z}}=5+10+10 \\
 & \Rightarrow {{P}_{x}}+{{P}_{y}}+{{P}_{z}}=25 \\
\end{align}\]
So the final answer to our question is 25 which is option a.
Option (a) is the correct option.

Note: In this type of question we can clearly see that the basic requirements to solve the question are the values of l, m and n and also the distance between the two points so always check whether both of these are mentioned in the question so as to move further with the question.