Determine all values of k for which each pair of vectors is orthogonal.
a) (1,2) and (k,k)
b) (1,2,1) and (k,2k,4)
To determine all values of k for which each pair of vectors is orthogonal, we need to check if the dot product of the vectors is equal to zero.
a) For the vectors (1,2) and (k,k), the dot product is calculated as:
(1,2) · (k,k) = 1*k + 2*k = k + 2k = 3k
For the vectors to be orthogonal, the dot product must be equal to zero:
3k = 0
Solving for k:
3k = 0
k = 0/3
k = 0
Therefore, for the vectors (1,2) and (k,k) to be orthogonal, k must equal 0.
b) For the vectors (1,2,1) and (k,2k,4), the dot product is calculated as:
(1,2,1) · (k,2k,4) = 1*k + 2*2k + 1*4 = k + 4k + 4 = 5k + 4
For the vectors to be orthogonal, the dot product must be equal to zero:
5k + 4 = 0
Solving for k:
5k = -4
k = -4/5
Therefore, for the vectors (1,2,1) and (k,2k,4) to be orthogonal, k must equal -4/5.