Question

The direction cosines of the line joining the points \[\left( {4,3,-{\text{ }}5} \right)\] and \[\left( {-2,1,-{\text{ }}8} \right)\] are
A. \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{2}{7}} \right)\left( {\dfrac{3}{7}} \right)\]
B. \[\left( {\dfrac{2}{7}} \right),\left( {\dfrac{3}{7}} \right)\left( {\dfrac{{ - 6}}{7}} \right)\]
C. \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{3}{7}} \right)\left( {\dfrac{2}{7}} \right)\]
D. None of these

Answer 1

Hint:
The angles that a line forms with the positive directions of the coordinate axes are known as the direction cosines of a line. Here we have given two points. Firstly we have to determine the DR’s of the line joining given points. To get the DC’s divide each DR by the distance between these two points.

Formula Used:
Direction Cosine of line joining two points whose coordinates are $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)=\left(\dfrac{x_2 - x_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{y_2 - y_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{z_2 - z_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}\right)$

Complete step-by-step answer:
Given There are two points given \[\left( {4,3,-{\text{ }}5} \right)\] and\[\left( {-2,1,-{\text{ }}8} \right)\].
Let these point be P\[\left( {4,3,-{\text{ }}5} \right)\]and Q\[\left( {-2,1,-{\text{ }}8} \right)\]
Direction ratio of PQ;
$=Position vector of P−Position vector of Q
=4−(−2),3−1,−5−(−8)
=6,2,3$
Therefore, Direction cosine of PQ;
$=\left(\dfrac{6}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{2}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{3)}{\sqrt {(-2 - 4)^2 + (1 - 3)^2 + (-8 + 5)^2}}\right)\\
=\left(\dfrac{6}{\sqrt {36+4+9}}, \dfrac{2}{\sqrt {36+4+9}}, \dfrac{3}{\sqrt {36+4+9}}\right)\\
=\left(\dfrac{6}{7}, \dfrac{2}{7}, \dfrac{3}{7}\right)$
Hence the correct option is A \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{2}{7}} \right)\left( {\dfrac{3}{7}} \right)\] .

So, option A is correct.

Note: To determine direction cosines, we use the correlation between direction ratios and cosines. Also need to know the formulas to find direction cosine and direction ratio of a line like Direction Cosine $ = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$