Question

If A, B, C, D are four distinct points in space such that $\overset{\rightarrow}{AB}$ is not perpendicular to $\overset{\rightarrow}{CD}$and satisfies $\overset{\rightarrow}{AB}$.$\overset{\rightarrow}{CD}$ = k (|$\overset{\rightarrow}{AD}$|² + |$\overset{\rightarrow}{BC}$|² – |$\overset{\rightarrow}{AC}$|² – |$\overset{\rightarrow}{BD}$|²) then the value of k is –

• A. 2
• B. 1/3
• C. ½
• D. 1

Answer 1

Answer: (D)

1

Answer 2

Let A is origin and position vectors of B, C and D are $\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$

$\overset{\rightarrow}{AB}$.$\overset{\rightarrow}{CD}$ = k(|$\overset{\rightarrow}{AD}$|² + |$\overset{\rightarrow}{BC}$|² – |$\overset{\rightarrow}{AC}$|² – |$\overset{\rightarrow}{BD}$|²)

̃ ($\overrightarrow{b}$).($\overrightarrow{d}$ – $\overrightarrow{c}$) = k (($\overrightarrow{d}$)² + ($\overrightarrow{c}$ – $\overrightarrow{b}$)² – ($\overrightarrow{c}$)²

– ($\overrightarrow{d}$– $\overrightarrow{b}$)²) k = ½