Question

The sum, difference and cross product of two vectors$\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$are mutually perpendicular if :

• A. $\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$are perpendicular to each other and |$\overset{\rightarrow}{A}$| = |$\overset{\rightarrow}{B}$|
• B. $\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$are perpendicular to each other
• C. $\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$are perpendicular but their magnitudes are arbitrary
• D. |$\overset{\rightarrow}{A}$| = |$\overset{\rightarrow}{B}$| and their directions are arbitrary

Answer 1

Answer: (D)

|$\overset{\rightarrow}{A}$| = |$\overset{\rightarrow}{B}$| and their directions are arbitrary

Answer 2

Let $\overset{\rightarrow}{A}$= a₁$\widehat{i}$+ a₂$\widehat{j}$+ a₃$\widehat{k}$and $\overset{\rightarrow}{B}$= b₁$\widehat{i}$+ b₂$\widehat{j}$+ b₃$\widehat{k}$

Given that $\overset{\rightarrow}{A}$+$\overset{\rightarrow}{B}$ is perpendicular to $\overset{\rightarrow}{A}$–$\overset{\rightarrow}{B}$

i.e., ($\overset{\rightarrow}{A}$+$\overset{\rightarrow}{B}$) . ($\overset{\rightarrow}{A}$–$\overset{\rightarrow}{B}$) = 0

or (a₁ + b₁) (a₁ – b₁) + (a₂ + b₂) (a₂ – b₂) + (a₃ + b₃) (a₃ – b₃) = 0

or a₁² + a₂² + a₃² = b₁² + b₂² + b₃²

or |$\overset{\rightarrow}{A}$| = |$\overset{\rightarrow}{B}$|

cross product of $\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$is perpendicular to the plane formed by $\overset{\rightarrow}{A}$and$\overset{\rightarrow}{B}$ or $\overset{\rightarrow}{A}$+$\overset{\rightarrow}{B}$ and $\overset{\rightarrow}{A}$–$\overset{\rightarrow}{B}$.