Show that the points A(1, 2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram. Show that ABCD is not a rectangle.
To prove: Points A, B, C, D form parallelogram
Formula: The distance between two points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ is given by
$\mathrm{D}=\sqrt{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}+\left(\mathrm{z}_{2}-\mathrm{z}_{1}\right)^{2}}$
Here
$\left(x_{1}, y_{1}, z_{1}\right)=(1,2,3)$
$\left(x_{2}, y_{2}, z_{2}\right)=(-1,-2,-1)$
$\left(x_{3}, y_{3}, z_{3}\right)=(2,3,2)$
$\left(x_{4}, y_{4}, z_{4}\right)=(4,7,6)$
Length $A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$
$=\sqrt{(-1-1)^{2}+(-2-2)^{2}+(-1-3)^{2}}$
$=\sqrt{(-2)^{2}+(-4)^{2}+(-4)^{2}}$
$=\sqrt{4+16+16}$
$=\sqrt{36}$
Length $B C=\sqrt{\left(x_{3}-x_{2}\right)^{2}+\left(y_{3}-y_{2}\right)^{2}+\left(z_{3}-z_{2}\right)^{2}}$
$=\sqrt{(2+1)^{2}+(3+2)^{2}+(2+1)^{2}}$
$=\sqrt{(3)^{2}+(5)^{2}+(3)^{2}}$
$=\sqrt{9+25+9}$
$=\sqrt{43}$
Length $C D=\sqrt{\left(x_{4}-x_{3}\right)^{2}+\left(y_{4}-y_{3}\right)^{2}+\left(z_{4}-z_{3}\right)^{2}}$
$=\sqrt{(4-2)^{2}+(7-3)^{2}+(6-2)^{2}}$
$=\sqrt{(2)^{2}+(4)^{2}+(4)^{2}}$
$=\sqrt{4+16+16}$
$=\sqrt{36}$
Length $A D=\sqrt{\left(x_{4}-x_{1}\right)^{2}+\left(y_{4}-y_{1}\right)^{2}+\left(z_{4}-z_{1}\right)^{2}}$
$=\sqrt{(4-1)^{2}+(7-2)^{2}+(6-3)^{2}}$
$=\sqrt{(3)^{2}+(5)^{2}+(3)^{2}}$
$=\sqrt{9+25+9}$
$=\sqrt{43}$
Length $A C=\sqrt{\left(x_{3}-x_{1}\right)^{2}+\left(y_{3}-y_{1}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}}$
$=\sqrt{(2-1)^{2}+(3-2)^{2}+(2-3)^{2}}$
$=\sqrt{(1)^{2}+(1)^{2}+(-1)^{2}}$
$=\sqrt{1+1+1}$
$=\sqrt{3}$
Length $\mathrm{BD}=\sqrt{\left(\mathrm{x}_{4}-\mathrm{x}_{2}\right)^{2}+\left(\mathrm{y}_{4}-\mathrm{y}_{2}\right)^{2}+\left(\mathrm{z}_{4}-\mathrm{z}_{2}\right)^{2}}$
$=\sqrt{(4+1)^{2}+(7+2)^{2}+(6+1)^{2}}$
$=\sqrt{(5)^{2}+(9)^{2}+(7)^{2}}$
$=\sqrt{25+81+49}$
$=\sqrt{155}$
Here, AB = CD which are opposite sides of polygon.
$B C=A D$ which are opposite sides of polygon.
Also the diagonals $A C$ and $B D$ are not equal in length.
Hence, the polygon is not a rectangle.
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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,