Question being asked: Use the trigonometric ratios and Pythagorean theorem to add the vectors given in the figure. Determine the magnitude and direction of the resultant.
The problem has two vectors, the first labeled A is pointing towards the origin in quadrant 1. With A=90.3 and the angle equaling 10.3 degrees.
The second b is pointing away from the origin with B=53.1 and the angle equaling 17.3 degrees.
So I know you move A so that the tail is at the origin. However after that I am not sure how to resolve the vectors into x and y components and all that comes after. Any help would be appreciated.
Yo are correct that you place both tails at the origin. Then for each vector v (that is, either A or B),
x = v cosθ
y = v sinθ
Remember your polar coordinates?
Then just add the x and y values to get the resultant.
Or, you can consider them as complex numbers, and that will take care of the x and y values for you
90.3 cis190.3° + 53.1 cis17.3° = 38.15 cis -3.13°
Or, geometrically, draw a parallelogram with sides A and B, and the diagonal will be the resultant.
Or, you can put the point of A at (0,0) and the tail of B at (0,0) and then A+B is just the vector from the tail of A to the tip of B.
review vector addition, or search online for many more examples.
To add the vectors A and B using trigonometric ratios and the Pythagorean theorem, follow these steps:
1. Draw vector A pointing towards the origin in quadrant 1. Label the magnitude of A as 90.3 and the angle it forms with the positive x-axis as 10.3 degrees. Draw vector B pointing away from the origin, with a magnitude of 53.1 and an angle of 17.3 degrees.
2. Move vector A so that its tail is at the origin. This step allows you to resolve the vectors into their x and y components more easily.
3. Resolve vector A into its x and y components using trigonometric ratios. The x-component (Ax) can be found by multiplying the magnitude of A (90.3) by the cosine of the angle (10.3 degrees). The y-component (Ay) can be found by multiplying the magnitude of A (90.3) by the sine of the angle (10.3 degrees). So, Ax = 90.3 * cos(10.3) and Ay = 90.3 * sin(10.3).
4. Resolve vector B into its x and y components using the same method. The x-component (Bx) can be found by multiplying the magnitude of B (53.1) by the cosine of the angle (17.3 degrees). The y-component (By) can be found by multiplying the magnitude of B (53.1) by the sine of the angle (17.3 degrees). So, Bx = 53.1 * cos(17.3) and By = 53.1 * sin(17.3).
5. Add the x-components and y-components separately to get the resultant vectors in the x and y directions. Rx = Ax + Bx, and Ry = Ay + By.
6. Use the Pythagorean theorem to find the magnitude of the resultant vector. The magnitude (R) can be found by taking the square root of the sum of the squares of the x and y components. R = sqrt(Rx^2 + Ry^2).
7. To determine the direction of the resultant vector, use trigonometry. The angle (θ) that the resultant vector makes with the positive x-axis can be found using the inverse tangent function. θ = tan^(-1)(Ry/Rx).
8. Convert the angle from radians to degrees if necessary.
So, by following these steps, you should be able to determine the magnitude and direction of the resultant vector.
To solve this problem, you can follow these steps:
1. Draw the vectors A and B on a coordinate plane, with the tail of vector A at the origin (0, 0).
2. To find the x and y components of vector A, you can use the trigonometric ratios. The x component (A_x) is given by A * cos(angle), and the y component (A_y) is given by A * sin(angle). Plugging in the values, A_x = 90.3 * cos(10.3°) and A_y = 90.3 * sin(10.3°).
3. Repeat step 2 for vector B to find its x and y components. B_x = 53.1 * cos(17.3°) and B_y = 53.1 * sin(17.3°).
4. Now, add the x components and the y components separately. A_x + B_x will give you the x component of the resultant vector, and A_y + B_y will give you the y component.
5. Use the Pythagorean theorem to find the magnitude (or length) of the resultant vector. The magnitude is given by sqrt((x component)^2 + (y component)^2).
6. To find the direction of the resultant vector, you can use the inverse tangent (or arctan) function. The direction (θ) is equal to arctan(y component / x component).
7. You now have the magnitude and direction of the resultant vector. Make sure to include the correct units and use the appropriate angle convention (e.g., clockwise or counterclockwise from the positive x-axis).
By following these steps, you should be able to solve the problem and find the magnitude and direction of the resultant vector.