Question

If the coordinates of the point A, B, C and D be $\left( {2,3, - 1} \right),\left( {3,5, - 3} \right),\left( {1,2,3} \right)$ and $\left( {3,5,7} \right)$ respectively, then the projection of $\overrightarrow {AB} $ on $\overrightarrow {CD} $ is
 A. $0$
 B. $1$
 C. $2$
 D. $3$

Answer 1

Hint: In order to solve this type of question, we will first find the direction ratios of $\overrightarrow {AB} $ and $\overrightarrow {CD} $ by substituting the values obtained. Next, we will find the dot product of $\overrightarrow {AB} $ and $\overrightarrow {CD} $ to find the projection of $\overrightarrow {AB} $ on $\overrightarrow {CD} $. Again, we will substitute the values obtained above to get the correct answer.

Formula used:
Direction ratios of the line passing through a line $A\left( {{x_1},{y_1},{z_1}} \right)$ and $B\left( {{x_2},{y_2},{z_2}} \right)$ is given by,
$\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$
$\overrightarrow {AB} .\overrightarrow {CD} = \left( {{a_2}} \right)\left( {{a_1}} \right) + \left( {{b_2}} \right)\left( {{b_1}} \right) + \left( {{c_2}} \right)\left( {{c_1}} \right)$

Complete step by step solution:
For $\overrightarrow {AB} $,
$A\left( {2,3, - 1} \right)$ and $B\left( {3,5, - 3} \right)$
Direction ratios of $\overrightarrow {AB} $,
$\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$
Substituting the values,
$\left( {3 - 2} \right),\left( {5 - 3} \right),\left( { - 3 - \left( { - 1} \right)} \right)$
$\therefore {a_1} = 1,\;{b_1} = 2,\;{c_1} = - 2$ ………………..equation $\left( 1 \right)$
$\overrightarrow {AB} = \;\widehat i + 2\widehat j - 2\widehat k$
For $\overrightarrow {CD} $,
$C\left( {1,2,3} \right)$ and $D\left( {3,5,7} \right)$
Direction ratios of $\overrightarrow {CD} $,
$\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$
Substituting the values,
$\left( {3 - 1} \right),\left( {5 - 2} \right),\left( {7 - 3} \right)$
$\therefore {a_2} = 2,\;{b_2} = 3,\;{c_2} = 4$ ………………..equation $\left( 2 \right)$
$\overrightarrow {CD} = 2\widehat i + 3\widehat j + 4\widehat k$
Now, we will find the dot product of $\overrightarrow {AB} $ and $\overrightarrow {CD} $,
$\overrightarrow {AB} .\overrightarrow {CD} = \left( {{a_2}} \right)\left( {{a_1}} \right) + \left( {{b_2}} \right)\left( {{b_1}} \right) + \left( {{c_2}} \right)\left( {{c_1}} \right)$
Substituting the values from equation $\left( 1 \right)$ and $\left( 2 \right)$
 $\overrightarrow {AB} .\overrightarrow {CD} = \left( 2 \right)\left( 1 \right) + \left( 3 \right)\left( 2 \right) - \left( 2 \right)\left( 4 \right)$
$\overrightarrow {AB} .\overrightarrow {CD} = 2 + 6 - 8$
$\overrightarrow {AB} .\overrightarrow {CD} = 0$
Thus, the projection of $\overrightarrow {AB} $ on $\overrightarrow {CD} $ is 0.
$\therefore $ The correct option is A.

Note: The direction ratios are very helpful in finding the relationship between two lines or vectors. The direction ratios can be used to find the direction cosines of a line or the angle between the two lines. The direction ratios are also useful in finding the dot product between the two vectors.