Question

The scalar product of the vector `veca=hati+hatj+hatk` with a unit vector along the sum of vectors `vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk` is equal to one. Find the value of λ and hence, find the unit vector along `vecb +vecc`

Answer 1

Given:

`veca=hati+hatj+hatk, vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk`

now 

`vecb+vecc=(2+lambda)hati+6hatj-2hatk`

Let `hatd` denote the unit vector along `vecb+vecc` Then,

`hatd=(vecb+vecc)/|vecb+vecc|`

`=>hatd=((2+lambda)hati+6hatj-2hatk)/sqrt((2+lambda)^2+(6)^2+(-2)^2)`

`=>hatd=((2+lambda)hati+6hatj-2hatk)/sqrt((2+lambda)^2+40)`

``

Also `veca.hatd=1`

`=>(hati+hatj+hatk).((2+lambda)hati+6hatj-2hatk)/sqrt((2+lambda)^2+40)=1`

`=>(hati+hatj+hatk)[(2+lambda)hati+6hatj-2hatk]=sqrt((2+lambda)^2+40)`

`=>2+lambda+6-2=sqrt(2+lambda)^2+40)`

`=>(lambda+6)^2=(2+lambda)^2+40`

`=>8lambda=8`

`=>lambda=1`

`therefore hatd=((2+lambda)hati+6hatj-2hatk)/sqrt((2+1)^2+40)=(3hati+6hatj-2hatk)/sqrt(49)`

`"i.e " hatd=1/7(3hati+6hatj-2hatk)`