Finding missing Scalars - Linear Algebra

Question

Hi , I have been very confused on this question . Your help is very very appreciated.

Consider the vectors

u = [ 0,1,1] and v = [ 2 , -2 , 1 ]

Let w = mv + k√10u * v

where m and k are elements of all real numbers

Find the condition on m and k that provides ∥w∥ ≤ √3

The answer must show a relation between m and k

Answer 1

u•v = I i j k I = 3i + 2j - 2k = [3, 2, -2]; w = [0, m, m] + √10[3k, 2k, -2k] = [3√10k, m + 2√10k, m - 2√10k]

I 0 1 1 I

I 2 -2 1]

IIwII = √(90k² + 2m² + 80k²) = √(170k² + 2m²) ≤ √3; 170k² + 2m² ≤ 3