The three displacement vectors have a magnitudes of a= 5.00 meters, b= 5.00 meters, and c= 4.00 meters. Find the resultant magnitude and Direction angle of the three vectors by names of the component method. Express the directional angle as an angle above the positive or negative x axis
To find the resultant magnitude and direction angle of the three displacement vectors using the component method, we need to break down each vector into its x and y components.
Let's assume that vector a is along the positive x-axis, vector b makes an angle of 60 degrees with the positive x-axis (in the counterclockwise direction), and vector c makes an angle of -30 degrees with the positive x-axis (in the clockwise direction).
Vector a can be represented as (5.00, 0) since it is purely along the x-axis.
To find the x and y components of vector b, we use trigonometry:
b_x = b * cos(angle)
b_y = b * sin(angle)
b_x = 5.00 * cos(60) ≈ 2.50
b_y = 5.00 * sin(60) ≈ 4.33
Similarly, for vector c:
c_x = c * cos(angle)
c_y = c * sin(angle)
c_x = 4.00 * cos(-30) ≈ 3.46
c_y = 4.00 * sin(-30) ≈ -2.00
Now, we can add the x and y components to find the resultant x and y components:
r_x = a_x + b_x + c_x
r_y = a_y + b_y + c_y
r_x = 5.00 + 2.50 + 3.46 ≈ 11.96
r_y = 0 + 4.33 - 2.00 ≈ 2.33
The resultant magnitude, r, can be found using the Pythagorean theorem:
r = sqrt(r_x^2 + r_y^2)
r = sqrt(11.96^2 + 2.33^2) ≈ 12.21 meters
To find the direction angle above the positive or negative x-axis, we can use the arctan function:
θ = arctan(r_y / r_x)
θ = arctan(2.33 / 11.96) ≈ 11.08 degrees
Therefore, the resultant magnitude is approximately 12.21 meters, and the direction angle is approximately 11.08 degrees above the positive x-axis.