Question

The projection of the line segment joining the points \[A\left( { - 1,{\text{ }}0,{\text{ }}3} \right)\] and \[B\left( {2,{\text{ }}5,{\text{ }}1} \right)\] on the line whose direction ratios are proportional to \[6,{\text{ }}2,{\text{ }}3\] is
A. $\dfrac{{10}}{7}$
B. $\dfrac{{22}}{7}$
C. $\dfrac{{18}}{7}$
D. None of these

Answer 1

Hint: We have to find projection of the line segment joining the points \[A\left( { - 1,{\text{ }}0,{\text{ }}3} \right)\] and \[B\left( {2,{\text{ }}5,{\text{ }}1} \right)\] on the line whose direction ratios are proportional to \[6,{\text{ }}2,{\text{ }}3\]. First, we will find the direction ratio of the line using given points. Then use formula Projection of line ${l_1}$ to ${l_2}$ $\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}$ for finding required solution.

Formula Used: Projection of line ${l_1}$ to ${l_2}$ =$\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}$

Complete Step by step solution: Given, the line segment joining the points \[A\left( { - 1,{\text{ }}0,{\text{ }}3} \right)\] and \[B\left( {2,{\text{ }}5,{\text{ }}1} \right)\]
Direction ratio of this line be $R(2 + 1,5 - 0,1 - 3)$
$R(3,5, - 2)$
Now, assign the values so that we can apply formulas.
Now, ${a_1} = 3,{b_1} = 5,{c_1} = - 2$
${a_2} = 6,{b_2} = 2,{c_2} = 3$
Projection of line ${l_1}$ to ${l_2}$ =$\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}$
After putting values
$ = \dfrac{{3 \times 6 + 5 \times 2 + 3 \times ( - 2)}}{{\sqrt {{6^2} + {2^2} + {3^2}} }}$
After solving
$ = \dfrac{{18 + 10 - 6}}{{\sqrt {36 + 4 + 9} }}$
$ = \dfrac{{22}}{{\sqrt {49} }}$
After simplification
$ = \dfrac{{22}}{7}$

Hence option B is correct.

Note: Students should concentrate while solving the question in order to avoid any mistakes. They should use the concept of direction ratio correctly to find the direction ratios of a given line. Use projection formula $\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_2^2 + b_2^2 + c_2^2} }}$ accurately to get exact required answer without any mistake or complications.