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

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$A$ ).