
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
162.9k+ views
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.
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.
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NEET 2025 – Every New Update You Need to Know

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

NEET Total Marks 2025

1 Billion in Rupees
