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)
just recall that orthogonal means that u•v = 0. So,
(a) (1,2)•(k,k) = k + 2k = 0, so k=0
(b) (1,2,1)•(k,2k,4) = k+4k+4 = 0, so k = -4/5
Kind of boring examples, IMO.
To determine if two vectors are orthogonal, we need the dot product of the vectors to be zero. Recall that the dot product of two vectors is given by the sum of the products of their corresponding components.
a) For the vectors (1,2) and (k,k), the dot product is given by:
(1,2) · (k,k) = 1*k + 2*k = k + 2k = 3k.
For the dot product to be zero, we need 3k = 0. Solving for k, we find:
k = 0.
Therefore, the vectors (1,2) and (k,k) are orthogonal for k = 0.
b) For the vectors (1,2,1) and (k,2k,4), the dot product is given by:
(1,2,1) · (k,2k,4) = 1*k + 2*2k + 1*4 = k + 4k + 4 = 5k + 4.
For the dot product to be zero, we need 5k + 4 = 0. Solving for k, we find:
k = -4/5.
Therefore, the vectors (1,2,1) and (k,2k,4) are orthogonal for k = -4/5.