18. The value of \( x \) for which angle between vectors \( 2 \hat{i}+x \hat{j}-3 \hat{k} \) and \( 4 \hat{i}+\hat{j}+\hat{k} \) is acute is/are (o) \( x-5 \) (c) \( x=-5 \) (d) none of these

Question

18. The value of \( x \) for which angle between vectors \( 2 \hat{i}+x \hat{j}-3 \hat{k} \) and \( 4 \hat{i}+\hat{j}+\hat{k} \) is acute is/are (o) \( x<-5 \) (b) \( x>-5 \) (c) \( x=-5 \) (d) none of these

Answer 1

A = 2i +xj - 3k

|A| =  sqrt (4 + x² + 9 ) = sqrt( 13 +x²)

B = 4i + j + k

|B| =  sqrt ( 16 + 1 + 1 ) = sqrt ( 18 )

Dot product of A and B = 8 + x -3 = 5 + x

5 + x = |A| |B| cosø

5 + x = sqrt ( 18 (13 +x²) ) cosø 

for cosø  to be positive  x + 5 > 0

then x > -5