Question

If OP = 21 and direction cosines of \[\overrightarrow{OP}\] are $\left( \dfrac{2}{7},\dfrac{6}{7},-\dfrac{3}{7} \right)$, then find the coordinates of P.
(a) (6, -12, 4)
(b) (6, 18, -9)
(c) $\left( \dfrac{3}{2},-\dfrac{6}{2} \right)$
(d) (5, -10, 6)

Answer 1

Hint: To solve this problem we need to have basic knowledge about vectors, direction ratios and direction cosines, so first of all direction cosines of a given vector is given by the ratio of the direction ratio of that vector to the magnitude of that vector. So we will use this to solve this problem. We will assume the coordinates of the point P as (x, y, z) and then equate given direction cosines with the mentioned formula and solve it to find the coordinates of P.

Complete step by step answer:
We are given that OP = 21, and
Direction cosines of \[\overrightarrow{OP}\] are $\left( \dfrac{2}{7},\dfrac{6}{7},-\dfrac{3}{7} \right)$
We know that according to the definition of the direction cosine, they are the ratio of direction ratio of the vector to the magnitude of that vector,
i.e.
$\operatorname{direction}\,cosine\,=\dfrac{diraction\,ratio}{magnitude}$
Now we will assume the coordinates of the point P as (x, y, z),
So we have,
$\begin{align}
  & \overrightarrow{OP}=\overrightarrow{P}-\overrightarrow{O}=\left( x,y,z \right)-\left( 0,0,0 \right) \\
 & \overrightarrow{OP}=\left( x,y,z \right) \\
\end{align}$
And magnitude of vector \[\overrightarrow{OP}\] is already given as 21,
So by comparing each direction cosine as following, we get
$\begin{align}
  & \dfrac{x}{21}=\dfrac{2}{7} \\
 & \Rightarrow x=\dfrac{2}{7}\times 21 \\
 & \Rightarrow x=6 \\
\end{align}$
Similarly, we get
$\begin{align}
  & \dfrac{y}{21}=\dfrac{6}{7} \\
 & \Rightarrow y=\dfrac{6}{7}\times 21 \\
 & \Rightarrow y=18 \\
\end{align}$
And,
$\begin{align}
  & \dfrac{z}{21}=-\dfrac{3}{7} \\
 & \Rightarrow z=-\dfrac{3}{7}\times 21 \\
 & \Rightarrow z=-9 \\
\end{align}$
Hence we get,
(x, y, z) = (6, 18, -9)

So, the correct answer is “Option B”.

Note: To solve these kinds of problems you need to have basic knowledge of the vectors so kindly go through the basic topics of the vectors. You can also solve this question by taking the general ratio of the given direction cosines and the option like general ratio of $\left( \dfrac{2}{7},\dfrac{6}{7},-\dfrac{3}{7} \right)$ is (2, 6, -3) and if we multiply this by 3, we will get option b, no other option has same general ratio.