Show that the following points are collinear :
$A(3,-5,1), B(-1,0,8)$ and $C(7,-10,-6)$
To prove: the 3 points are collinear.
Formula: The distance between two points $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{Z}_{1}\right)$ and $\left(\mathrm{x}_{2}, \mathrm{y}_{2}, \mathrm{Z}_{2}\right)$ is given by
$\mathrm{D}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$
Here,
$\left(x_{1}, y_{1}, z_{1}\right)=(3,-5,1)$
$\left(x_{2}, y_{2}, z_{2}\right)=(-1,0,8)$
$\left(x_{3}, y_{3}, z_{3}\right)=(7,-10,-6)$
Length $\mathrm{AB}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$
$=\sqrt{(-1-3)^{2}+(0+5)^{2}+(8-1)^{2}}$
$=\sqrt{(-4)^{2}+(5)^{2}+(7)^{2}}$
$=\sqrt{16+25+49}$
$=\sqrt{90}=3 \sqrt{10}$
Length $\mathrm{BC}=\sqrt{\left(x_{3}-x_{2}\right)^{2}+\left(y_{3}-y_{2}\right)^{2}+\left(z_{3}-z_{2}\right)^{2}}$
$=\sqrt{(7+1)^{2}+(-10-0)^{2}+(-6-8)^{2}}$
$=\sqrt{(8)^{2}+(-10)^{2}+(-14)^{2}}$
$=\sqrt{64+100+196}$
$=\sqrt{360}=6 \sqrt{10}$
Length $\mathrm{AC}=\sqrt{\left(x_{3}-x_{1}\right)^{2}+\left(y_{3}-y_{1}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}}$
$=\sqrt{(7-3)^{2}+(-10+5)^{2}+(-6-1)^{2}}$
$=\sqrt{(4)^{2}+(-5)^{2}+(-7)^{2}}$
$=\sqrt{16+25+49}$
$=\sqrt{90}=3 \sqrt{10}$
$B A+B C=3 \sqrt{10}+3 \sqrt{10}=6 \sqrt{10}=B C$
Therefore $A, B, C$ are collinear.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,