Vector A has a magnitude 2.4m and direction 130 degrees relative to the positive x-axis. Vector B has magnitude 3.0m and direction 250 degrees relative to the positive x-axis.
1) The resultant R=A+B has magnitude(in m) ?
I know the R =2.75m but I am not sure how to get this. I know you have to use sqrt of rx^2 + ry^2 but I am not sure how to get Ry.
You can simply compute the squared norm of R as follows. Writing the squared norm of a vector X as X^2 (which is also the inner product of X with itself, so this notation makes sense):
R^2 = (A + B)^2 =
A^2 + B^2 + 2 A dot B.
A dot B = |A| |B| cos(angle between A and B)
The angle is 250° - 130° = 120°
Cos(120°) = Cos(90° + 30°) = -Sin(30°) = -1/2
So:
|R| = sqrt[A^2 + B^2 - |A||B|] =
sqrt[7.56] = 2.74
why is sin negative?
You can use the identity:
Sin(x)= Cos(90° - x)
So, if you have Cos(90° + x), you can replace x by -x in the above identity:
Cos(90° + x) = Sin(-x) = -Sin(x)
Note that Sin is an odd function so it changes sign when the sign of x is changed:
Sin(-x) = -Sin(x)
Cos is an even function:
Cos(-x) = Cos(x)
To find the magnitude of the resultant vector R, you need to break down the vectors A and B into their x and y components.
Let's start with vector A. The magnitude of vector A is given as 2.4 m, and the direction is 130 degrees relative to the positive x-axis. To find the x and y components of A, you can use trigonometry.
The x component of A, denoted as Ax, can be found using the cosine function:
Ax = A * cos(θ), where A is the magnitude and θ is the angle.
Substituting the given values:
Ax = 2.4 m * cos(130°)
Similarly, the y component of A, denoted as Ay, can be found using the sine function:
Ay = A * sin(θ), where A is the magnitude and θ is the angle.
Substituting the given values:
Ay = 2.4 m * sin(130°)
Repeat the same process for vector B using its magnitude of 3.0 m and direction of 250 degrees relative to the positive x-axis.
Once you have the x and y components for both vectors, you can add their x components together and their y components together. This will give you the x and y components of the resultant vector R.
Finally, you can use the Pythagorean theorem to find the magnitude of R:
R = sqrt(Rx^2 + Ry^2)
Substituting the calculated values for Rx and Ry, you can find the magnitude of R. In the given case, you mentioned that R = 2.75 m, so you would need to double-check your calculations or clarify if the provided result is correct.