Copy of Vector A has a magnitude of 9 and is at an angle of 30.4 degrees measured counter-clockwise from the x axis. Vector B has a magnitude of 12 and is at an angle of 102.3 degrees measured counter-clockwise from the x axis. Vector C has a magnitude of 18 and is at an angle of 180 degrees measured counter-clockwise from the x axis. Lastly, Vector D has a magnitude of 16 and is at an angle of 293.8 degrees measured counter-clockwise from the x axis. If R = A + B + C - D, find the magnitude of R.
R = 9[30.4o] + 12[102.3o] + 18[180o] +
16[293.8o].
X = 9*Cos30.4+12*Cos102.3+18*Cos180+
16*Cos293.8 =
Y = 9*sin30.4+12*sin102.3+18*sin180+
16*sin293.8 =
R = sqrt(X^2 + Y^2).
To find the magnitude of vector R, which is the sum of vectors A, B, C, and the negation of vector D, we need to break each vector into its horizontal and vertical components. Then we can add up the horizontal and vertical components separately and find the magnitude of the resultant vector using the Pythagorean theorem.
Let's start by finding the horizontal and vertical components for each vector:
For vector A:
Magnitude = 9
Angle = 30.4 degrees measured counter-clockwise from the x-axis.
The horizontal component (Ax) can be found using the formula:
Ax = Magnitude * cos(Angle)
Ax = 9 * cos(30.4 degrees)
For vector B:
Magnitude = 12
Angle = 102.3 degrees measured counter-clockwise from the x-axis.
The horizontal component (Bx) can be found using the formula:
Bx = Magnitude * cos(Angle)
Bx = 12 * cos(102.3 degrees)
For vector C:
Magnitude = 18
Angle = 180 degrees measured counter-clockwise from the x-axis.
The horizontal component (Cx) can be found using the formula:
Cx = Magnitude * cos(Angle)
Cx = 18 * cos(180 degrees)
For vector D:
Magnitude = 16
Angle = 293.8 degrees measured counter-clockwise from the x-axis.
The horizontal component (Dx) can be found using the formula:
Dx = Magnitude * cos(Angle)
Dx = 16 * cos(293.8 degrees)
Now let's add up the horizontal components:
Rx = Ax + Bx + Cx - Dx
Similarly, we can find the vertical components for each vector using the formula:
Ay = Magnitude * sin(Angle)
By = Magnitude * sin(Angle)
Cy = Magnitude * sin(Angle)
Dy = Magnitude * sin(Angle)
Then add up the vertical components:
Ry = Ay + By + Cy - Dy
Once we have the horizontal component (Rx) and vertical component (Ry), we can find the magnitude of vector R using the Pythagorean theorem:
Magnitude of R = sqrt((Rx)^2 + (Ry)^2)
Substitute the calculated values and evaluate the expression to find the magnitude of vector R.