The scalar product of the vector ` hat i+\ hat j+\ hat k\ ` with the unit vector along the sum of vectors `2 hat i+\ 4 hat j-5 hat k` . and `lambda hat i+\ 2 hat j+\ 3 hat k\ ` is equal to one. Find the value of `lambda` .

Answer

1

Explanation

Let `vec(a) = 2hat(i) + 4hat(j)- 5hat(k) and vec(b)= lamda hat(i)+ 2hat(j) + 3hat(k)` Then `vec(a)+ vec(b) = (2 + lamda) hat(i) + 6hat(j)- 2hat(k)`
Unit vector along `vec(a) + vec(b)= ((2 + lamda) hat(i) + 6hat(j)- 2hat(k))/(sqrt((2+ lamda)^(2) +6^(2)+ (-2)^(2)))`
According to given condition, `(hat(i)+ hat(j) + hat(k)). ((2 + lamda) hat(i)+ 6hat(j)- 2hat(k))/(sqrt((2+ lamda)^(2)+6^(2)+ (-2)^(2)))=1`
`rArr 1 xx (2+ lamda) +1 xx 6 +1 xx (-2)= sqrt((2+ lamda)^(2)+ 40)`
`rArr lamada +6= sqrt((2+ lamda)^(2)+ 40)`
`rArr lamda^(2)+ 12 lamda + 36 = 4+4 lamda+ lamda^(2)+ 40`
`rArr 12 lamda + 36= 4 lamda + 44 rArr lamda =1`