If `vec(a)=hat(i)-2hat(j)+5hat(k) and vec(b)=2hat(i)+hat(j)-3hat(k)` then what is `(vec(b)-vec(a)). (3vec(a)+vec(b))` equal to ?

Options

A. 106
B. -106
C. 53
D. -53

Answer

B

Explanation

Given, `vec(a)=hat(i)-2hat(j)+5hat(k)`
and `vec(b)=2hat(i)+hat(j)-3hat(k)`
`:.vec(b)-vec(a)=2hat(i)+hat(j)-3hat(k)-hat(i)+2hat(j)-5hat(k)=hat(i)+3hat(j)-8hat(k)`
and `(3vec(a)+vec(b))=(3hat(i)-6hat(j)+15hat(k))+(2hat(i)+hat(j)-3hat(k))`
`=5hat(i)-5hat(j)+12hat(k)`
Hence, `(vec(b)-vec(a)).(3vec(a)+vec(b))=(hat(i)+3hat(j)-8hat(k)).(5hat(i)-5hat(j)+12hat(k))`
`=5-15-96`
`=-106`