Displacement vectors
A,
B,
and
C
add up to a total of zero. Vector
A
has a magnitude of 1550 m and a direction of 22.9° north of east. Vector
B
has a direction of 41.0° east of south, and vector
C
has a direction of 35.2° north of west. Find the magnitudes of vector
B
and vector
C.
magnitude of
B
m
magnitude of
C
m
To find the magnitudes of vectors B and C, we can make use of the fact that the sum of the vectors A, B, and C is zero. This means that the magnitudes of A, B, and C form a right triangle.
First, let's break down vector A into its horizontal and vertical components. The horizontal component (Ax) is given by:
Ax = magnitude of A * cos(angle of A)
Substituting the values, we get:
Ax = 1550 m * cos(22.9°)
Next, the vertical component (Ay) of vector A is given by:
Ay = magnitude of A * sin(angle of A)
Substituting the values, we get:
Ay = 1550 m * sin(22.9°)
Now, let's analyze the horizontal and vertical components of the given vectors B and C to find their magnitudes.
For vector B, the horizontal component (Bx) is given by:
Bx = magnitude of B * cos(angle of B)
Substituting the values, we get:
Bx = magnitude of B * cos(41.0°)
Similarly, the vertical component (By) of vector B is given by:
By = magnitude of B * sin(angle of B)
Substituting the values, we get:
By = magnitude of B * sin(41.0°)
For vector C, the horizontal component (Cx) is given by:
Cx = magnitude of C * cos(angle of C)
Substituting the values, we get:
Cx = magnitude of C * cos(35.2°)
Similarly, the vertical component (Cy) of vector C is given by:
Cy = magnitude of C * sin(angle of C)
Substituting the values, we get:
Cy = magnitude of C * sin(35.2°)
Now, since the sum of A, B, and C is zero:
Ax + Bx + Cx = 0 (horizontal components add up to zero)
Ay + By + Cy = 0 (vertical components add up to zero)
We can use these equations to solve for the magnitudes of B and C.
By rearranging the above equations, we get:
magnitude of B = ( -Ax - Cx ) / cos(41.0°)
magnitude of C = ( -Ay - By ) / sin(35.2°)
Substituting the calculated values of Ax, Cx, Ay, and By, we can find the magnitudes of B and C.