Why is my resultant values different in my graphical and calculated method?

Question

R = A + B - C

A = 30.0m, 30 degrees

B = 40.0m, 125 degrees

C = 35.0m, 300 degrees

Why does my calculated resultant length value way off than my graphical. My graphical resultant measures about 77 m but my calculated resultant measures around 22 m. Which is correct? I think it's because of that vector "-C".

Rx = Ax + Bx - Cx

= 25cos30 + 40cos125 - 35cos120

= 21.7 + (-22.9) - (-17.5)

= 16.7

Ry = Ay + By - Cy

= 25sin30 + 40sin125 - 35sin120

= 12.5 + (45.3) - (30.3)

= 15.0

^^^Is there something wrong with that calculation which explains why my resultant values are different? If its not the calculation, then perhaps my graphical method. I might have drawn it incorrect. Otherwise, please tell me which value is correct, 77 or 22?

Answer 1

So, you are solving for the resultant vector right.

In this case the problem is two-dimension, thus we expect that the resultant vector will be dependent on x and y ( horizontal and vertical components)

In Vector Calculus the formula for solving for the resultant is

R' = A' + B' + C'

   where R', A', B', C' are all vectors having x and y components.

Since the different vectors are all lying in the same origin we can compute for the resultant vector using the component method.

X - component:

The x-component of the resultant vector can be solved using summation.

Rx = Ax + Bx + Cx

      = 30 cos30 + 40 cos125 + 35cos300

      = 25.98 + (-22.94) + 17.5

Rx = 20.54 m

Y - component:

Again the y-component of the resultant vector can be solved using summation.

Ry = Ay + By + Cy

      = 30 sin30 + 40 sin125 + 35 sin300

      = 15 + 32.77 + (-30.3)

Ry = 17.46

To solve for the magnitude of the resultant vector we need to use the Pythagorean theorem.

R = Sqrt [  Rx ^ 2   +  Ry ^ 2  ]

    = Sqrt [ 421.89  +  304.82  ]

    = Sqrt [ 726.71 ]

    = 26.96 m

To solve for the angle we will use the inverse tangent function.

  angle = arctan [ Ry / Rx ]

           = arctan [ 17.46 / 20.54 ]

           = arctan [ 0.85 ]

  angle = 40.37 degrees

As you can see your calculated value(22m) and your graphical value(77m) were both incorrect.

The value that you get from your graphical solution can be immediately seen to be incorrect since it is way too large compare to the other 3 vectors. Since the 3 vectors are dispersed in different directions you can't expect to have an answer way more than their value. And your calculated result was also incorrect because you write vector C to be negative in the formula which is wrong since the vector sum is always positive. Let the trigonometric function do the computation whether it is positive or negative.  

Answer 2

C is at angle 300 degrees with x, then why did you use cos 120? Why not cos 300? If you want to convert to angle less than 180, then note that cos 300 is not equal to cos 120. Cos 300 = +0.5 and cos 120 = -0.5. One is positive and another is negative.

Cos 300 = Cos(360 - 60) = Cos(-60) = Cos(60) = 0.5

Sin 300 = Sin(360 - 60) = Sin(-60) = -Sin(60) = -0.86

Rx = Ax + Bx - Cx

= 25cos30 + 40cos125 - 35cos300

= 21.7 + (-22.9) - (17.5)

= -18.7

Ry = Ay + By - Cy

= 25sin30 + 40sin125 - 35sin300

= 12.5 + (45.3) - (-30.3)

= 88.1

Magnitude of resultant vector R = sqrt(Rx^2 + Ry^2)

= sqrt[(-18.7)^2 + 88.1^2]

= sqrt(8111.3)

= 90.1

For graphical method, assume a vector D such that

D = -C

D = 35.0m, +60 degrees (C makes 300 i.e. -60 degrees. Therefore D makes +60 degrees)

Now

R = A + B + D

Show A, B, D in the graph.