Final answer:
Two vectors are parallel if their corresponding components have equal ratios. The given vectors are parallel and can be expressed as scalar multiples of each other. Adding a scalar multiple of one vector to the other does not result in a vector that is a scalar multiple of either original vector.
Step-by-step explanation:
Two vectors are parallel if they point in the same direction or in opposite directions. This means that if two vectors are parallel, one must be a scalar multiple of the other.
To determine if two vectors are parallel, we can compare their components. If the ratios of the corresponding components are equal, then the vectors are parallel. Let's apply this to the given vectors:
(a) -(-2,4) and v = (9,6,-12)
Comparing the components, we have:
-2/9 = 4/6 = -1/2
The ratios are equal, so the vectors are parallel. To express v as a scalar multiple of u, we divide each component of v by its corresponding component in u:
u = 2(9,6,-12)
(b) u = (-9,-6,12) and v = (12,8,-16)
Comparing the components, we have:
-9/12 = -6/8 = 12/-16 = -3/4
The ratios are equal, so the vectors are parallel. To express u as a scalar multiple of v, we divide each component of u by its corresponding component in v:
u = -4(12,8,-16)
(c) Given u = (6,2), v = (1,2), and k = 2
Adding the vectors, we have:
u + kv = (6,2) + 2(1,2)
Distributing the scalar, we get:
u + kv = (6,2) + (2,4)
Adding the corresponding components, we have:
u + kv = (6+2, 2+4) = (8,6)
Since u + kv is not a scalar multiple of either u or v, the vectors are not parallel.