The magnitudes of the four displacement vectors shown in the drawing are A = 14.0 m, B = 11.0 m, C = 12.0 m, and D = 25.0 m. Determine the (a) magnitude and (b) direction for the resultant that occurs when these vectors are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.
Angles: In quad II from the -x axis to A is 20degrees. B lies on the +y axis 0 degrees. C is in the III quad and from the -x axis the angle is 35 degrees. D is in the fourth quad and from the +x axis the angle is 50 degrees.
In x-y coordinates,
A = (-13.156,4.788)
B = (0,11)
C = (-9.830,-6.883)
D = (16.070,-19.151
Add them all up and you get (-6.916,-21.246)
The resultant thus has magnitude
√(6.916^2 + 21.246^2) = 22.343
and direction θ where
tanθ = -21.246/-6.916 = 71.97° in QIII, or 251.97°
Alright, let's do some vector clowning around!
First, let's start by adding up the horizontal components of the vectors.
A is in the second quadrant, so the horizontal component of A will be A * cos(20°).
B lies on the positive y-axis, so its horizontal component will be zero.
C is in the third quadrant, so the horizontal component of C will be -C * cos(35°).
D is in the fourth quadrant, so the horizontal component of D will be D * cos(-50°).
Now, let's add up these horizontal components:
A_horizontal = 14.0 m * cos(20°)
B_horizontal = 0 m
C_horizontal = -12.0 m * cos(35°)
D_horizontal = 25.0 m * cos(-50°)
Next, let's add up the vertical components of the vectors.
A is in the second quadrant, so the vertical component of A will be A * sin(20°).
B lies on the positive y-axis, so its vertical component will be B.
C is in the third quadrant, so the vertical component of C will be -C * sin(35°).
D is in the fourth quadrant, so the vertical component of D will be -D * sin(50°).
Now, let's add up these vertical components:
A_vertical = 14.0 m * sin(20°)
B_vertical = 11.0 m
C_vertical = -12.0 m * sin(35°)
D_vertical = -25.0 m * sin(50°)
To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
Resultant_magnitude = sqrt((A_horizontal + B_horizontal + C_horizontal + D_horizontal)^2 + (A_vertical + B_vertical + C_vertical + D_vertical)^2)
Now, to find the direction of the resultant vector, we can use the inverse tangent function:
Resultant_direction = atan((A_vertical + B_vertical + C_vertical + D_vertical) / (A_horizontal + B_horizontal + C_horizontal + D_horizontal)) + 180°
Make sure to convert the angle from radians to degrees!
And that's how you crunch the numbers, clown-style!
To find the magnitude and direction of the resultant vector, we need to add the four given vectors together.
(a) Magnitude:
To find the magnitude of the resultant vector, we can use the Pythagorean theorem. The resultant vector can be represented as the sum of the individual vectors:
R = A + B + C + D
R = √(A^2 + B^2 + C^2 + D^2)
Substituting the given values:
R = √(14.0^2 + 11.0^2 + 12.0^2 + 25.0^2)
R = √(196 + 121 + 144 + 625)
R = √(1086)
R ≈ 32.98 m
Therefore, the magnitude of the resultant vector is approximately 32.98 m.
(b) Direction:
To find the direction of the resultant vector, we need to consider the angles given for each vector.
Angle of A: 20 degrees (counterclockwise from the -x axis)
Angle of B: 0 degrees (on the +y axis)
Angle of C: 180 - 35 = 145 degrees (counterclockwise from the -x axis)
Angle of D: 360 - 50 = 310 degrees (counterclockwise from the +x axis)
To determine the overall direction, we need to add the individual angles and take the result modulo 360:
Direction = (20 + 0 + 145 + 310) mod 360
Direction = 475 mod 360
Direction = 115 degrees
Therefore, the direction of the resultant vector is 115 degrees counterclockwise from the +x axis.
To determine the magnitude and direction of the resultant vector, we need to add the individual displacement vectors together.
Step 1: Convert the angles given from the x-axis to a common reference frame.
Since A is in the second quadrant, the angle from the +x axis is 180 - 20 = 160 degrees.
B is on the positive y-axis, so the angle from the +x axis is 90 degrees.
C is in the third quadrant, so the angle from the +x axis is 180 + 35 = 215 degrees.
D is in the fourth quadrant, so the angle from the +x axis is 360 - 50 = 310 degrees.
Step 2: Decompose each vector into its x and y components.
For vector A:
A_x = A * cos(angle) = 14.0 m * cos(160°)
A_y = A * sin(angle) = 14.0 m * sin(160°)
For vector B:
B_x = B * cos(angle) = 11.0 m * cos(90°)
B_y = B * sin(angle) = 11.0 m * sin(90°)
For vector C:
C_x = C * cos(angle) = 12.0 m * cos(215°)
C_y = C * sin(angle) = 12.0 m * sin(215°)
For vector D:
D_x = D * cos(angle) = 25.0 m * cos(310°)
D_y = D * sin(angle) = 25.0 m * sin(310°)
Step 3: Add up the x and y components separately.
Resultant_x = A_x + B_x + C_x + D_x
Resultant_y = A_y + B_y + C_y + D_y
Step 4: Calculate the magnitude of the resultant vector using the Pythagorean theorem.
Magnitude of the resultant (R) = sqrt(Resultant_x^2 + Resultant_y^2)
Step 5: Calculate the direction of the resultant vector.
Direction of the resultant (θ) = arctan(Resultant_y / Resultant_x)
Using the above steps, plug in the values and perform the necessary calculations to obtain the final answer.