Question

The vectors \((\overset{⃗}{A}+\overset{⃗}{B})\) and \((\overset{⃗}{A}-\overset{⃗}{B})\) are perpendicular to each other. This is possible under the condition

• A. \(|\overset{⃗}{A}| = |\overset{⃗}{B}|\)
• B. \(\overset{⃗}{A}\cdot \overset{⃗}{B} = 0\)
• C. \(\overset{⃗}{A}\times \overset{⃗}{B} = 0\)
• D. \(\overset{⃗}{A}\cdot \overset{⃗}{B} = 1\)

Hint:

Set the dot product (A + B) . (A - B) equal to zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Two vectors are perpendicular when their dot product is zero.

Step 2: Expand:
\((\vec A+\vec B)\cdot(\vec A-\vec B)=\vec A\cdot\vec A-\vec A\cdot\vec B+\vec B\cdot\vec A-\vec B\cdot\vec B=|\vec A|^2-|\vec B|^2\).

Step 3: Set to zero:
\(|\vec A|^2-|\vec B|^2=0\), so \(|\vec A|=|\vec B|\). Option A.

Step 4: Why the other options are wrong.
\(\vec A\cdot\vec B=0\) means A is perpendicular to B, which is a different condition. \(\vec A\times\vec B=0\) means parallel vectors. \(\vec A\cdot\vec B=1\) has no link to the condition.

Final Answer:
A and B must have equal magnitudes. \[ \boxed{\text{(A) }|\vec A|=|\vec B|} \]