To find the unit vector in the direction of vector c, we start with the given vectors: a = (-1, 3) and b = (4, 7). The vector c is defined as:

c = 4a + 2b.

First, we calculate vector c by scaling the components of vectors a and b:

4a = 4 * (-1, 3) = (-4, 12)

2b = 2 * (4, 7) = (8, 14)

Now, we add the corresponding components:

c = (-4 + 8, 12 + 14) = (4, 26).

Next, we find the magnitude of vector c using the formula for the magnitude of a vector:

|c| = √(x² + y²),

where x and y are the components of vector c. Plugging in the values:

|c| = √(4² + 26²) = √(16 + 676) = √(692).

To simplify √(692), we factor it as:

√(692) = √(4 * 173) = √(4) * √(173) = 2√(173).

Now, we can find the unit vector c hat (𝛩) by dividing vector c by its magnitude:

𝛩 = c / |c| = (4, 26) / (2√(173)).

This can be expressed as:

𝛩 = (4 / (2√(173)), 26 / (2√(173))) = (2 / √(173), 13 / √(173)).

To rationalize the denominators, we multiply the numerator and denominator by √(173):

𝛩 = (2√(173) / 173, 13√(173) / 173).

This gives us the final unit vector in the direction of c:

𝛩 = (2√(173) / 173, 13√(173) / 173).

By following these steps—calculating vector c, finding its magnitude, and then deriving the unit vector—you can effectively solve similar problems involving vectors and their directions.