
The direction cosines of the line joining the points \[\left( {4,3,-{\text{ }}5} \right)\] and \[\left( {-2,1,-{\text{ }}8} \right)\] are
A. \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{2}{7}} \right)\left( {\dfrac{3}{7}} \right)\]
B. \[\left( {\dfrac{2}{7}} \right),\left( {\dfrac{3}{7}} \right)\left( {\dfrac{{ - 6}}{7}} \right)\]
C. \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{3}{7}} \right)\left( {\dfrac{2}{7}} \right)\]
D. None of these
Answer
164.1k+ views
Hint:
The angles that a line forms with the positive directions of the coordinate axes are known as the direction cosines of a line. Here we have given two points. Firstly we have to determine the DR’s of the line joining given points. To get the DC’s divide each DR by the distance between these two points.
Formula Used:
Direction Cosine of line joining two points whose coordinates are $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)=\left(\dfrac{x_2 - x_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{y_2 - y_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{z_2 - z_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}\right)$
Complete step-by-step answer:
Given There are two points given \[\left( {4,3,-{\text{ }}5} \right)\] and\[\left( {-2,1,-{\text{ }}8} \right)\].
Let these point be P\[\left( {4,3,-{\text{ }}5} \right)\]and Q\[\left( {-2,1,-{\text{ }}8} \right)\]
Direction ratio of PQ;
$=Position vector of P−Position vector of Q
=4−(−2),3−1,−5−(−8)
=6,2,3$
Therefore, Direction cosine of PQ;
$=\left(\dfrac{6}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{2}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{3)}{\sqrt {(-2 - 4)^2 + (1 - 3)^2 + (-8 + 5)^2}}\right)\\
=\left(\dfrac{6}{\sqrt {36+4+9}}, \dfrac{2}{\sqrt {36+4+9}}, \dfrac{3}{\sqrt {36+4+9}}\right)\\
=\left(\dfrac{6}{7}, \dfrac{2}{7}, \dfrac{3}{7}\right)$
Hence the correct option is A \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{2}{7}} \right)\left( {\dfrac{3}{7}} \right)\] .
So, option A is correct.
Note: To determine direction cosines, we use the correlation between direction ratios and cosines. Also need to know the formulas to find direction cosine and direction ratio of a line like Direction Cosine $ = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$
The angles that a line forms with the positive directions of the coordinate axes are known as the direction cosines of a line. Here we have given two points. Firstly we have to determine the DR’s of the line joining given points. To get the DC’s divide each DR by the distance between these two points.
Formula Used:
Direction Cosine of line joining two points whose coordinates are $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)=\left(\dfrac{x_2 - x_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{y_2 - y_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}, \dfrac{z_2 - z_1}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2- z_1)^2}}\right)$
Complete step-by-step answer:
Given There are two points given \[\left( {4,3,-{\text{ }}5} \right)\] and\[\left( {-2,1,-{\text{ }}8} \right)\].
Let these point be P\[\left( {4,3,-{\text{ }}5} \right)\]and Q\[\left( {-2,1,-{\text{ }}8} \right)\]
Direction ratio of PQ;
$=Position vector of P−Position vector of Q
=4−(−2),3−1,−5−(−8)
=6,2,3$
Therefore, Direction cosine of PQ;
$=\left(\dfrac{6}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{2}{\sqrt {(-2 - 4)^2 + (1 + 3)^2 + (-8 + 5)^2}}, \dfrac{3)}{\sqrt {(-2 - 4)^2 + (1 - 3)^2 + (-8 + 5)^2}}\right)\\
=\left(\dfrac{6}{\sqrt {36+4+9}}, \dfrac{2}{\sqrt {36+4+9}}, \dfrac{3}{\sqrt {36+4+9}}\right)\\
=\left(\dfrac{6}{7}, \dfrac{2}{7}, \dfrac{3}{7}\right)$
Hence the correct option is A \[\left( {\dfrac{6}{7}} \right),\left( {\dfrac{2}{7}} \right)\left( {\dfrac{3}{7}} \right)\] .
So, option A is correct.
Note: To determine direction cosines, we use the correlation between direction ratios and cosines. Also need to know the formulas to find direction cosine and direction ratio of a line like Direction Cosine $ = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

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

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

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

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

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

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Advanced 2025 Notes

JEE Main Chemistry Question Paper with Answer Keys and Solutions
