Vector B is 3.46 units long toward the east. Vector A is 4.00 units long 30 degrees N of E. How long and in what direction is C = A - B?
C=A-B=A+(-B)
Resolve A and -B into the x (due east) and y (due north) components and add to get the resultant C.
C = A - B ,
C = 4 @ 30 Deg - 3.46 @ 0 deg,
X = Hor = 4*cos30 - 3.46 = 3.46 - 3.46 = 0,
Y = Ver = 4*sin30 - 0 = 2,
C = 0 + i2 = 2 @ 90 deg N of E.
To find the magnitude and direction of vector C = A - B, we need to subtract the components of vector B from the components of vector A.
Given:
Magnitude of vector B = 3.46 units (toward the east)
Magnitude of vector A = 4.00 units
Angle of vector A with respect to east is 30 degrees N of E.
Step 1: Resolve vector B into its east and north components.
Since vector B is directed solely toward the east, its east component will be equal to its magnitude, and the north component will be zero.
The east component of vector B = 3.46 units.
The north component of vector B = 0 units.
Step 2: Resolve vector A into its east and north components.
To find the east and north components of vector A, we'll use trigonometry.
The angle of vector A with respect to east is 30 degrees N of E. This means the angle between vector A and the positive x-axis is 30 degrees.
The east component of vector A = magnitude of vector A * cos(angle)
The north component of vector A = magnitude of vector A * sin(angle)
The east component of vector A = 4.00 units * cos(30 degrees)
The east component of vector A = 4.00 units * √3/2
The east component of vector A ≈ 3.464 units
The north component of vector A = 4.00 units * sin(30 degrees)
The north component of vector A = 4.00 units * 1/2
The north component of vector A = 2.00 units
So, the east component of vector A is approximately 3.464 units, and the north component is 2.00 units.
Step 3: Subtract the east and north components of vector B from the east and north components of vector A to get the components of vector C.
The east component of vector C = east component of vector A - east component of vector B
The east component of vector C = 3.464 units - 3.46 units
The east component of vector C ≈ 0.004 units
The north component of vector C = north component of vector A - north component of vector B
The north component of vector C = 2.00 units - 0 units
The north component of vector C = 2.00 units
Step 4: Calculate the magnitude and direction of vector C using its components.
The magnitude of vector C = √(east component of vector C)^2 + (north component of vector C)^2
The magnitude of vector C = √(0.004 units)^2 + (2.00 units)^2
The magnitude of vector C ≈ √0.000016 units^2 + 4.00 units^2
The magnitude of vector C ≈ √16.000016 units^2
The magnitude of vector C ≈ 4.000 units
The direction of vector C can be found by measuring the angle it makes with the positive x-axis (east direction). We'll use the inverse tangent function to calculate this angle.
Angle of vector C = arctan(north component of vector C / east component of vector C)
Angle of vector C = arctan(2.00 units / 0.004 units)
Angle of vector C ≈ arctan(500)
Using a calculator, the approximate angle of vector C is approximately 88.725 degrees.
Therefore, the magnitude of vector C is approximately 4.000 units, and its direction is approximately 88.725 degrees from the east direction.