The vector ` vec c `, directed along the internal bisector of the angle between the vectors `vec a = 7 hati - 4 hatj - 4hatk and vecb = -2hati - hatj + 2 hatk " with " |vec c| = 5 sqrt(6),` is

Options

A. `(5)/(3)(hati -7hatj + 2hatk)`
B. `(5)/(3)(5hati +5hatj + 2hatk)`
C. `(5)/(3)(hati +7hatj + 2hatk)`
D. `(5)/(3)(-5hati +5hatj + 2hatk)`

Answer

A

Explanation

The required vector `vec c` is given by
`vec c =lambda (hata + hat b)`
`rArr vec c = lambda ((veca)/(|veca|)+(vecb)/(|vecb|))`
`rArr vec c = lambda .(1)/(9) (7hati - 4 hatj - 4 hatk ) + (1)/(3) (-2hati -hatj + 2 hatk ) }`
`rArr vec c = (lambda )/(9) (hati - 7 hatj + 2hatk)`
`rArr |vec c|= pm (lambda)/(9) sqrt(1+49+4)=pm (lambda)/(9) sqrt(54)`
But ` | vec c| = 5 sqrt(6) " "`[Given]
` rArr pm (lambda)/(9)sqrt(54) = 5 sqrt(6) rArr lambda= pm 15`
Hence, `vec c = pm (15)/(9) (hati - 7 hati + 2 hatk ) = pm (5)/(3) (hati -7hatj + 2 hatk)`