Final answer:
To find the constants a, b, and c, the components of the given vectors are compared. After the comparison, it is determined that a = 3, b = 9, and c = -1.5.
Step-by-step explanation:
The given vectors are u = i + 3j - 4k and v = ai + ck. It's also given that u * v = 3i + bj - 6k. Assuming '*' denotes the dot product, the product u * v will result in a scalar, not a vector, which seems to be a typo in the question. However, if '*' denotes a direct element-wise product or Hadamard product, then the individual components of vectors u and v would multiply to give the respective components of u * v.
Comparing the components of u * v and 3i + bj - 6k:
- a (from ai) must be 3 since i component of u is 1.
- b (from bj) must be 9 because j component of u is 3 and to get bj as a result, b has to be 3 * 3 = 9.
- c (from ck) must be -1.5 since k component of u is -4 and to obtain -6k, -4 * c must equal -6.
Hence, a = 3, b = 9, and c = -1.5.