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 $( \lambda )$