Given that p = (4i - 3j) and q = (-i + 5j), find r such that |r| = 15 and is in the direction of (2p + 3q). A. 2i + j B. 3i + 7j C. -4i + 11j D. 6i - 9j

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asked Jul 14, 2024 131k views

1 Answer

Matthew Abbott · Answer 1 · 2024-07-21T08:32:20+0000

There is no vector r that satisfies both the magnitude and direction requirements.

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.