Question

If a x b and c x d are perpendicular satisfying a.c = λ, b.d = λ (λ>0) and a.d = 4, b.c = 9, then λ is equal to:

Answer 1

Given that vectors a and b are perpendicular, as well as vectors c and d.

From this, we can conclude:
\([ a \cdot b = 0 ]\) and \(( c \cdot d = 0 )\) because the dot product of two perpendicular vectors is zero. 

We're also given: 
\(1) ( a \cdot c = \lambda )\)
\(2) ( b \cdot d = \lambda )\)
\(3) ( a \cdot d = 4 )\)
\(4) ( b \cdot c = 9 )\)

From the properties of the dot product: 
\([ (a + b) \cdot (c + d) = a \cdot c + a \cdot d + b \cdot c + b cdo\) 

Given that \(( a \times b )\) and\(( c \times d )\) are perpendicular, the vectors ( a + b ) and ( c + d ) are also perpendicular to each other. 

Thus: \([ (a + b) \cdot (c + d) = 0 ]\)

Substituting in the known dot products: 
\([ \lambda + 4 + 9 + \lambda = 0 ]\)
\([ 2\lambda + 13 = 0 ]\)
\([ 2\lambda = -13 ]\)
\([ \lambda = -\frac{13}{2} ]\) 

This result seems contradictory because we're given \(( \lambda > 0 ).\) It's possible that the problem may have an error or it might require additional context or constraints to yield a positive value for \(( \lambda )\)