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?
To find the magnitude and direction of vector C = A - B, we need to subtract vector B from vector A.
First, let's find the components of vector A and vector B:
Vector A has a magnitude of 4.00 units and is 30 degrees north of east (NE).
To find the x-component (east component) of A, we multiply the magnitude by the cosine of the angle:
Ax = 4.00 units * cos(30 degrees) = 3.46 units (approximately)
To find the y-component (north component) of A, we multiply the magnitude by the sine of the angle:
Ay = 4.00 units * sin(30 degrees) = 2.00 units
Vector B has a magnitude of 3.46 units towards the east.
Therefore, the x-component (east component) of B is 3.46 units, and the y-component (north component) of B is 0 units since it is purely towards the east.
Now, let's subtract the x-components and y-components to find the components of C:
Cx = Ax - Bx = 3.46 units - 3.46 units = 0 units
Cy = Ay - By = 2.00 units - 0 units = 2.00 units
The x-component of C is 0 units, and the y-component of C is 2.00 units.
To find the magnitude (length) of vector C, we can use the Pythagorean theorem:
|C| = sqrt(Cx^2 + Cy^2) = sqrt((0 units)^2 + (2.00 units)^2) = sqrt(0 units + 4.00 units^2) = 2.00 units
The magnitude of vector C is 2.00 units.
To find the direction of vector C, we can use the inverse tangent (arctan) function:
θ = atan(Cy / Cx) = atan(2.00 units / 0 units)
Since the x-component of C is 0 units, dividing by 0 units is undefined. Therefore, we cannot determine the exact direction of vector C. However, we can say that vector C is purely in the y-direction (north direction) with a magnitude of 2.00 units.