What are the x and y components of a vector a in the xy plane if it's direction is 215 degrees counterclockwise from the positive direction of the x axis and its magnitude is 12.9 m ?
you would be in the third quadrant, so both x and y are negative.
The values would have to satisfy two equations:
x^2 + y^2 = 12.9^2 and y/x = tan 35º
They are relatively easy to solve, I got x appr -10.567 and y appr. -7.399
To find the x and y components of a vector in the xy plane, given its direction and magnitude, you can use trigonometry.
1. Start by drawing a diagram representing the xy plane.
2. Locate the positive x-axis and measure the angle counterclockwise from that axis to the direction of the vector. In this case, the angle is 215 degrees counterclockwise from the positive x-axis.
3. Identify the quadrant in which the angle lies. Since 215 degrees is greater than 180 degrees but less than 270 degrees, it lies in the third quadrant.
4. In the third quadrant, both the x and y components of the vector will be negative.
5. Use the magnitude of the vector (12.9 m) and the known angle (215 degrees) to set up some trigonometric equations.
6. First, use the magnitude and the angle to find the ratio of the opposite side (y) to the adjacent side (x) in a right triangle. The tangent function can be used for this: tan(angle) = y/x. In this case, tan(215 degrees) = y/x.
7. Rearrange the equation to solve for y/x: y/x = tan(215 degrees).
8. Next, use the Pythagorean theorem to relate the magnitudes of the x and y components. The theorem states that the square of the hypotenuse (magnitude of the vector) equals the sum of the squares of the other two sides. In this case, x^2 + y^2 = (12.9 m)^2.
9. Now, you have two equations: y/x = tan(215 degrees) and x^2 + y^2 = 12.9^2. Solve these equations simultaneously to find the values of x and y.
10. Use a calculator or math software to evaluate the trigonometric function and solve for y/x. In this case, tan(215 degrees) is approximately -0.700207538.
11. Substitute this value into the first equation and rearrange it to solve for y: y = x * tan(215 degrees). Since both x and y are negative in the third quadrant, you can consider the absolute value of the tangent.
12. Plug in the values for y/x and solve for y: y = |x| * 0.700207538.
13. Substitute this expression for y in the second equation and solve for x: x^2 + (|x| * 0.700207538)^2 = 12.9^2.
14. Solve the equation for x by using numerical methods or approximations. In this case, x is approximately -10.567 m.
15. Substitute this value of x into the equation y = |x| * 0.700207538 to solve for y. In this case, y is approximately -7.399 m.
Therefore, the x and y components of the vector with a direction of 215 degrees counterclockwise from the positive x-axis and a magnitude of 12.9 m are approximately -10.567 m and -7.399 m, respectively.