Finding the x and y components of a vector given the magnitude and direction is a common problem in physics and engineering. Here's how you can do it step by step:
Let's say the magnitude of the vector is and the direction of the vector is given by the angle with respect to the positive x-axis.
1. To find the x-component of the vector, you can use the formula: The x-component is the magnitude of the vector times the cosine of the direction angle.
2. To find the y-component of the vector, you can use the formula: The y-component is the magnitude of the vector times the sine of the direction angle.
So, these are the simple steps you can follow to find the x and y components of a vector given the magnitude and direction.
If you don't understand why we use cosine and sine for finding the x and y components, let me explain. When you have a vector with a certain direction, you can think of it as having a horizontal component (x) and a vertical component (y). The cosine of the angle gives you the ratio of the adjacent side to the hypotenuse in a right-angled triangle, which corresponds to the x-component. Similarly, the sine of the angle gives you the ratio of the opposite side to the hypotenuse, which corresponds to the y-component.