# The scalar product of the vector a=i+j+k with a unit vector along the sum of vectors b=2i+4j−5k and c=λi+2j+3k is equal to one. Find the value of λ and hence, find the unit vector along b +c - Mathematics

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

#### Solution

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)

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Is there an error in this question or solution?