Question
If the sum and difference of two vector $\vec{A}$ and $\vec{B}$ are perpendicular to each other then prove that two vectors are equal in magnitude.
✓
Answer
Sum of two vectors $\vec{A}$ and $\vec{B}=\vec{A}+\vec{B}$
$\text { Difference }=\vec{A}-\vec{B}$
$\because$ It is given that both are perpendicular hence their scalar product will be zero.
$\begin{aligned}
(\vec{A}+\vec{B}) \cdot(\vec{A}-\vec{B}) & =0 \\
\text { or } \quad \vec{A} \cdot \vec{A}-\vec{A} \cdot \vec{B}+\vec{B} \cdot \vec{A}-\vec{B} \cdot \vec{B} & =0
\end{aligned}$
We know that $\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}$
$\begin{aligned}
A^2-B^2 & =0 \\
A & =B
\end{aligned}$
Hence, both vectors are equal in magnitude.
$\text { Difference }=\vec{A}-\vec{B}$
$\because$ It is given that both are perpendicular hence their scalar product will be zero.
$\begin{aligned}
(\vec{A}+\vec{B}) \cdot(\vec{A}-\vec{B}) & =0 \\
\text { or } \quad \vec{A} \cdot \vec{A}-\vec{A} \cdot \vec{B}+\vec{B} \cdot \vec{A}-\vec{B} \cdot \vec{B} & =0
\end{aligned}$
We know that $\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}$
$\begin{aligned}
A^2-B^2 & =0 \\
A & =B
\end{aligned}$
Hence, both vectors are equal in magnitude.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
What are thermal radiations? State their important properties.
A boy puts a heavy box of mass M on his head and jumps down from the top of a multistoried building to the ground. How much is the force exerted by the box on his head during his free fall? Does the force greatly increase during the period he balances himself after striking the ground?
Figure shows a circular wheel of radius 10.0cm whose upper half, shown dark in the figure, is made of iron and the lower half of wood. The two junctions are joined by an iron rod. A uniform magnetic field B of magnitude $2.00 \times 10^{-4} T$ exists in the space above the central line as suggested by the figure. The wheel is set into pure rolling on the horizontal surface. If it takes 2.00 seconds for the iron part to come down and the wooden part to go up, find the average emf induced during this period.
→Define gravitational potential energy. Give its SI unit.
Two vessels of different materials are identical in size and wall thickness. They are filled with equal quantities of ice at 0°C. If the ice melts completely in 10 and 25 min respectively, compare the coefficients of thermal conductivity of the materials of the vessels.
What are the limitations of the first law of thermodynamics?
A sample of 0.177 g of an ideal gas occupies $1000 \mathrm{~cm}^3$ at STP. Calculate the rms speed of the gas molecules.
→The moment of inertia of two rotating bodies $A$ and $B$ are $I_A$, and $I_B,\left(I_A>I_B\right)$ and their angular momentum are equal. Which of them has greater kinetic energy?
→It is said that a liquid rises or is depressed in a capillary due to the surface tension. If a liquid neither rises nor depresses in a capillary, can we conclude that the surface tension of the liquid is zero?
The density of water is greater than the density of air, yet the cloud containing small drops of water does not fall but keeps floating in the sky. Why?