Vector A is directed along the positive y-axis and has magnitude 5.0 units. Vector B is directed along the negative x-axis and has magnitude 6.0 unit.
(a) What are the magnitude and direction of A + B?
(b) What are the magnitude and direction of A − B?
(c) What are the x- and y-components of B − A?
To find the magnitude and direction of vector A + B, we need to first determine the components of vectors A and B using their magnitudes and directions.
Given:
Magnitude of vector A = 5.0 units (directed along the positive y-axis)
Magnitude of vector B = 6.0 units (directed along the negative x-axis)
(a) Vector A + B:
To find the magnitude, we add the magnitudes of A and B:
Magnitude of A + B = magnitude of A + magnitude of B = 5.0 + 6.0 = 11.0 units
To find the direction of A + B, we need to determine the angle it forms with the positive x-axis. We can use trigonometry to find this angle.
The x-component of A is 0 since it is directed along the y-axis. The y-component of A is 5.0 units.
The x-component of B is -6.0 units since it is directed along the negative x-axis. The y-component of B is 0 since it is directed along the y-axis.
To determine the angle, we can use the arctan function:
θ = arctan(y-component / x-component)
θ = arctan(5.0 / -6.0)
θ ≈ -40.56 degrees
Therefore, the magnitude of A + B is 11.0 units, and it is directed at an angle of approximately -40.56 degrees with respect to the positive x-axis.
(b) Vector A - B:
To find the magnitude, we subtract the magnitudes of A and B:
Magnitude of A - B = magnitude of A - magnitude of B = 5.0 - 6.0 = -1.0 units (magnitude is always positive)
To find the direction of A - B, we again need to determine the angle it forms with the positive x-axis.
The x-component of A is 0, and the y-component of A is 5.0 units.
The x-component of B is -6.0 units, and the y-component of B is 0.
θ = arctan(y-component / x-component) = arctan(5.0 / 6.0)
θ ≈ 40.56 degrees (in the fourth quadrant)
Therefore, the magnitude of A - B is 1.0 unit, and it is directed at an angle of approximately 40.56 degrees with respect to the positive x-axis.
(c) To determine the x- and y-components of B - A, we subtract the x- and y-components of A and B individually.
The x-component of B - A = x-component of B - x-component of A = -6.0 - 0 = -6.0 units
The y-component of B - A = y-component of B - y-component of A = 0 - 5.0 = -5.0 units
Therefore, the x-component of B - A is -6.0 units, and the y-component of B - A is -5.0 units.