Final answer:
To find vector b such that compab = 3 with vector a = [4, 0, -1], we can let b = k[4, 0, -1], determine k using the formula for component of b along a, and solve for k to get b = (3/√17)[4, 0, -1].
Step-by-step explanation:
The question requires finding a vector b such that the component of b along vector a is 3. The component of b along a (denoted as compab) can be found by the formula:
compab = (b · a) / |a|
where 'b · a' is the dot product of vectors b and a, and '|a|' is the magnitude of vector a.
To find such a vector b, we can choose its direction to be the same as that of vector a and adjust its magnitude to satisfy the equation compab = 3. Given that vector a = [4, 0, -1], the magnitude of a is |a| = √(42 + 02 + (-1)2) = √(16 + 1) = √17.
Now, we must have (b · a) / √17 = 3. Multiplying both sides by √17 gives us:
b · a = 3√17
Therefore, vector b can take the form of b = k[4, 0, -1], where k is a scalar. To find k, we use the dot product b · a = k(a · a) = k|a|2 = k(17). Setting k(17) = 3√17 gives us k = 3/√17. So,
vector b = (3/√17)[4, 0, -1]