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What are the different cosines of the joins of the following pairs of points
$\left( {6,3,2} \right){\text{ }}\left( {5,1,4} \right)$ ?
  A.{\text{ }}\dfrac{1}{3},{\text{ }}\dfrac{2}{{3{\text{ }}}},{\text{ }}\dfrac{2}{3} \\
  B.{\text{ }}\dfrac{2}{3},{\text{ }}\dfrac{1}{3},{\text{ - }}\dfrac{2}{3} \\
  C.{\text{ }}\dfrac{1}{3},{\text{ }}\dfrac{2}{3},{\text{ }}\dfrac{2}{3} \\
  D.{\text{ }}\dfrac{2}{3},{\text{ }}\dfrac{1}{3},{\text{ }}\dfrac{2}{3} \\

Last updated date: 15th Jul 2024
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Hint: In order to solve this question we will use the general formula of direction cosines. i.e
$\dfrac{{{x_2} - {x_1}}}{{AB}}{\text{ , }}\dfrac{{{y_2} - {y_1}}}{{AB}}{\text{ , }}\dfrac{{{z_2} - {z_1}}}{{AB}}$ where, $A$ and$B$ are the points of direction cosines.
Direction cosines, in analytical geometry the direction cosines of a vector are cosines of the angles between the vector and the three coordinate axes.

Complete step-by-step answer:
In this question we have to find the direction cosines of the points $\left( {6,3,2} \right){\text{ }}\left( {5,1,4} \right)$
Suppose that the direction cosines of the given points are $A$ and$B$ .
For$A\left( {{x_1},{y_1},{z_1}} \right)$ and for $B\left( {{x_2},{y_2},{z_2}} \right)$
Direction cosines is $\dfrac{{{x_2} - {x_1}}}{{AB}}{\text{ , }}\dfrac{{{y_2} - {y_1}}}{{AB}}{\text{ , }}\dfrac{{{z_2} - {z_1}}}{{AB}}$ (equation 1)
So for $A\left( {6,3,2} \right){\text{ , }}B\left( {5,1,4} \right)$
Put the value of$AB{\text{ }}$ into (equation1):
=$\dfrac{{6 - 5}}{3},{\text{ }}\dfrac{{3 - 1}}{2},{\text{ }}\dfrac{{2 - 4}}{3}$
=$\dfrac{1}{3},{\text{ }}\dfrac{2}{3},{\text{ }}\dfrac{2}{3}$
So, the right answer is $\dfrac{1}{3},{\text{ }}\dfrac{2}{3},{\text{ }}\dfrac{2}{3}$ i.e.
(Option$A$ ).

Note: Whenever we face such types of questions the key concept is that we should know the general formula of direction cosines. Here, in this question we simply apply the formula of direction cosines when the points are given.