Let `alpha in R` and the three vectors
`a=alpha hat(i) + hat(j) +3hat(k) , b=2hat(i) +hat(j) -alpha hat(k)`
`" and " c= alpha hat(i) -2hat(j) +3hat(k)` . Then the set
`S={alpha: a,b " and c are coplanar"}`

Options

A. is singleton
B. is empty
C. contains exactly twol positive numbers
D. contains exactly two numbers only one of which is positive

Answer

B

Explanation

Given the vectors are
` a= alpha hat(i) + hat(j) + 3hat(k)`
` b= 2hat(i) + hat(j) - alpha hat(k)`
`" and " c= alpha hat(i) - 2hat(j) + 3hat(k)`
Clearly `[a,b,c] = |{:(alpha,,1,,3),(2,,1,,-alpha),(alpha,,-2,,3):}|`
`=alpha (3 -2 alpha) - 1 (6+ alpha^(2)) + 3 (-4 -alpha)`
`=- 3alpha^(2) -18 =- 3 (alpha^(2) +6)`
`:' ` There is no value of `alpha` for which `-3(alpha^(2)+6)` becomes zero so = `|{:(alpha ,,1,,3),(2,,1,,-alpha),(alpha,,-2,,3):}|[a,b,c] ne 0`
`rArr` vectors a,b and c are not coplanar for any value `a in R`
So the set `={alpha : a,b` and c are coplanar} is empty set .