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.
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
I think your unit vector is
1/√(λ^2 + 2λ + 10) <1+λ, 3, 0>
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
To find the value of λ, we need to compute the scalar product of the vector - i - j - k with the unit vector along the sum of vectors i + 2j + k and λi + j - k.
First, let's calculate the unit vector along the sum of the two given vectors:
i + 2j + k + λi + j - k = (1 + λ)i + (2 + 1)j + (1 - 1)k
= (1 + λ)i + 3j
To obtain a unit vector, we need to divide this vector by its magnitude. The magnitude of a vector v = (v1, v2, v3) is given by:
|v| = √(v1² + v2² + v3²)
In this case, the magnitude of the sum of the two given vectors is:
|sum of vectors| = √((1 + λ)² + 3²)
To find the unit vector, we divide the sum of the vectors by its magnitude:
unit vector = ((1 + λ)i + 3j) / |sum of vectors|
Now, we can calculate the scalar product of the vector - i - j - k with the unit vector:
(-i - j - k) · unit vector
= (-1)(1 + λ)/ |sum of vectors| + (-1)(3)/ |sum of vectors|
Since the scalar product is equal to 1/2, we set the expression equal to 1/2:
(-1)(1 + λ)/ |sum of vectors| + (-1)(3)/ |sum of vectors| = 1/2
Simplifying the equation:
-(1 + λ)/ |sum of vectors| - 3/ |sum of vectors| = 1/2
Now, we can solve this equation for λ. We will proceed step by step:
1. Multiply both sides of the equation by |sum of vectors|:
-(1 + λ) - 3 = 1/2 * |sum of vectors|
2. Distribute the negative sign:
-1 - λ - 3 = (1/2) * |sum of vectors|
3. Combine like terms:
-4 - λ = (1/2) * |sum of vectors|
4. Multiply both sides of the equation by -1:
4 + λ = - (1/2) * |sum of vectors|
5. Subtract 4 from both sides of the equation:
λ = -4 - (1/2) * |sum of vectors|
Finally, substitute the magnitude of the sum of the vectors back into the equation to find the value of λ.