Question
The projections of a vector on the three coordinate axis are $$6,\,- 3,\, 2$$ respectively. The direction cosines of the vector are :
A.
$$\frac{6}{5},\,\frac{{ - 3}}{5},\,\frac{2}{5}$$
B.
$$\frac{6}{7},\,\frac{{ - 3}}{7},\,\frac{2}{7}$$
C.
$$\frac{{ - 6}}{7},\,\frac{{ - 3}}{7},\,\frac{2}{7}$$
D.
$$6,\, - 3,\,2$$
Answer :
$$\frac{6}{7},\,\frac{{ - 3}}{7},\,\frac{2}{7}$$
Solution :
Let $$P\left( {{x_1},\,{y_1},\,{z_1}} \right)$$ and $$Q\left( {{x_2},\,{y_2},\,{z_2}} \right)$$ be the initial and final points of the vector whose projections on the three coordinate axes are $$6,\, - 3,\,2$$
then
$${x_2} - {x_1} = 6\,;\,\,\,{y_2} - {y_1} = - 3\,;\,\,\,{z_2} - \,{z_1} = 2$$
So that direction ratios of $$\overrightarrow {PQ} $$ are $$6,\, - 3,\,2$$
$$\therefore $$ Direction cosines of $$\overrightarrow {PQ} $$ are
$$\eqalign{
& \frac{6}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},\,\frac{{ - 3}}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},\,\frac{2}{{\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }} \cr
& = \frac{6}{7},\,\frac{{ - 3}}{7},\,\frac{2}{7} \cr} $$