JEE MAIN - Mathematics (2009 - No. 21)
The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vector are :
$${6 \over 5},{{ - 3} \over 5},{2 \over 5}$$
$${6 \over 7 },{{ - 3} \over 7},{2 \over 7}$$
$${- 6 \over 7 },{{ - 3} \over 7},{2 \over 7}$$
$$6, -3, 2$$
Explanation
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 coordinates axes are $${6, - 3,2}$$ then
$${x_2} - {x_1}, = 6;\,\,{y_2} - {y_1} = - 3;\,\,{z_2} - {z_1} = 2$$
So that directions ratios of $$\overrightarrow {PQ} $$ are $${6, - 3,2}$$
$$\therefore$$ Direction cosines of $$\overrightarrow {PQ} $$ are
$${6 \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},{{ - 3} \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},$$
$$\,\,\,\,\,\,\,\,$$ $${2 \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }} = {6 \over 7},{{ - 3} \over 7},{2 \over 7}$$
$${x_2} - {x_1}, = 6;\,\,{y_2} - {y_1} = - 3;\,\,{z_2} - {z_1} = 2$$
So that directions ratios of $$\overrightarrow {PQ} $$ are $${6, - 3,2}$$
$$\therefore$$ Direction cosines of $$\overrightarrow {PQ} $$ are
$${6 \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},{{ - 3} \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }},$$
$$\,\,\,\,\,\,\,\,$$ $${2 \over {\sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {2^2}} }} = {6 \over 7},{{ - 3} \over 7},{2 \over 7}$$
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