Question

The scalar product of the vector - i- j - k with a unit vector along the sum of vectors i + 2j + k and λi+ j - k is equal to 1/2. Find the value of λ.

Answer 1

first, let's find the sum of vectors i + 2j + k and λi+ j - k
= <1,2,1) + < λ, 1, -1> , (using the standard <..., ..., ...> vector notation)
= < 1+λ, 3, 0>
a unit vector along that is √(λ^2 + 2λ + 10)<1+λ, 3, 0>

now do a scalar product of that with <-1,-1,-1>
= √(λ^2 + 2λ + 10)(-1-λ - 3 + 0) = 1/2
let's replace λ with x for easier typing
√(x^2 + 2x + 10)(-1-x - 3 + 0) = 1/2
square both sides
(x^2 + 2x + 10)(x^2 + 8x + 16) = 1/4

yukkk!!!!
Wolfram says, x or λ = -4.1156 or -3.8798
since I squared my equation, all answers must be checked
only λ = -4.1156 works

Was expecting "easier" answer, better check my calculations

Answer 2

I think your unit vector is
1/√(λ^2 + 2λ + 10) <1+λ, 3, 0>

Answer 3

Thanks for the catch, oobleck
I even wrote it out first on paper, and I did have that,
made the error when I typed it.
Arggghhhh!!!!

So we would be solving:
1/(x^2 + 2x + 10)(x^2 + 8x + 16) = 1/4
4x^2 + 32x + 64 = x^2 + 2x + 10
3x^2 + 30x + 54 = 0
x = √7 - 5 or x = -√7 - 5 , replace the x with λ

I checked both in
1/√(x^2 + 2x + 10)(-1-x - 3 + 0) = 1/2 , they both worked