Find the sum of two vectors if one has a direction of 37° and a magnitude of 15 and the other has a direction of 217° and a magnitude of 9.
A)magnitude of 24 with direction of 217°
B)magnitude of 6 with direction of 217°
C)magnitude of 6 with direction of 37°
D)magnitude of 24 with direction of 37°

Question

Quizplus · Accepted Answer

To find the sum of two vectors, we break each vector into its horizontal (x) and vertical (y) components. For the first vector (magnitude 15, direction 37°), its components are: $x_1 = 15\cos(37°)$ and $y_1 = 15\sin(37°)$. For the second vector (magnitude 9, direction 217°), its components are: $x_2 = 9\cos(217°)$ and $y_2 = 9\sin(217°)$. The sum of the vectors is the sum of their components: $x_{total} = x_1 + x_2$ and $y_{total} = y_1 + y_2$. The magnitude of the resultant vector is $\sqrt{x_{total}^2 + y_{total}^2}$ and its direction (angle) can be found using $\tan^{-1}(y_{total}/x_{total})$. Without calculating the exact numbers, the key insight is that the vectors are not directly opposite to each other, and thus, their sum will not have the magnitudes or directions as given in options A, B, and D. The correct magnitude and direction depend on the resultant vector's calculation, which typically would not revert back to the exact direction of one of the original vectors unless they are directly opposite, which in this case, they are not. The explanation provided here is to clarify the process, but the correct choice is identified based on understanding vector addition principles rather than the specific numerical outcome, which requires calculation.

Find the Sum of Two Vectors If One Has a Direction