Question

The direction cosine of a line which is perpendicular to both the lines whose direction ratios are $(1, - 2, - 2)$ and $(0,2,1)$ are
A. $(\dfrac{2}{3}, - \dfrac{1}{3},\dfrac{2}{3})$
B. $(\dfrac{2}{3},\dfrac{1}{3},\dfrac{2}{3})$
C. $(\dfrac{2}{3},\dfrac{1}{3}, - \dfrac{2}{3})$
D. $( - \dfrac{2}{3},\dfrac{1}{3},\dfrac{2}{3})$

Answer 1

To find out the direction cosines of a line perpendicular to the lines whose direction ratios are $(1, - 2, - 2)$ and $(0,2,1)$, we find the cross product of $(1, - 2, - 2)$ and $(0,2,1)$. This will give us the direction ratio of the line required, now we use $\dfrac{l}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{m}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{n}{{\sqrt {{l^2} + {m^2} + {n^2}} }}$ to find the direction cosines of the required line.

Complete step by step answer:

Given direction ratios are $(1, - 2, - 2)$ and $(0,2,1)$
The vectors that can be formed by given direction ratios are ,
$\widehat i - 2\widehat j - 2\widehat k$and $2\widehat j + \widehat k$
Now, we need to calculate the vector perpendicular to both the vectors so it can be given as ,

= \left| {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ 1&{ - 2}&{ - 2} \\\ 0&2&1 \end{array}} \right| \\\ = ( - 2 + 4)\widehat i - (1 - 0)\widehat j + (2 - 0)\widehat k \\\ = 2\widehat i - \widehat j + 2\widehat k \\\

Hence, the direction ratio of vector perpendicular to both the given direction ratios is $(2, - 1,2)$
Hence, the direction cosines are
$\dfrac{l}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{m}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{n}{{\sqrt {{l^2} + {m^2} + {n^2}} }}$

$$= \dfrac{2}{{\sqrt {{2^2} + {{( - 1)}^2} + {2^2}} }},\dfrac{{ - 1}}{{\sqrt {{2^2} + {{( - 1)}^2} + {2^2}} }},\dfrac{2}{{\sqrt {{2^2} + {{( - 1)}^2} + {2^2}} }} \\\ = \dfrac{2}{3},\dfrac{{ - 1}}{3},\dfrac{2}{3} \\\$$

So, option (a) is our required correct answer.

Note: Any number proportional to the direction cosine is known as the direction ratio of a line. These direction numbers are represented by a, b and c. We can conclude that some of the squares of the direction cosines of a line is $1$.
Once, direction ratios are known you can use the formula as per $\dfrac{l}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{m}{{\sqrt {{l^2} + {m^2} + {n^2}} }},\dfrac{n}{{\sqrt {{l^2} + {m^2} + {n^2}} }}$to calculate given direction cosines. Calculate without any mistake.