Final answer:
There is no vector r that satisfies both the magnitude and direction requirements.
Step-by-step explanation:
To find the vector r in the direction of (2p + 3q) and with a magnitude of 15, we can first compute (2p + 3q).
Given that p = (4i - 3j) and q = (-i + 5j), we can substitute these values into the expression:
(2p + 3q) = 2(4i - 3j) + 3(-i + 5j)
Simplifying this expression, we get:
(2p + 3q) = (8i - 6j) + (-3i + 15j)
(2p + 3q) = 5i + 9j
Now that we have the direction of the vector, we can find the unit vector in this direction by dividing the vector by its magnitude:
Unit vector in the direction of (2p + 3q) = (5i + 9j) / |5i + 9j|
To find the magnitude of the vector, we can use the formula:
|r| = 15
Now we can substitute the values into the equation and solve for r:
15 = |5i + 9j|
15 = √(5² + 9²)
15 = √(25 + 81)
15 = √106
15 = 10.295
Since the two sides of the equation are not equal, we can conclude that there is no vector r that satisfies both conditions.