Given the following vectors:
A = (28, 110°)
B = (63, -30°)
Determine A + B. Specify the sum as both Cartesian and polar.
given the angles are measured from the x axis
A=28(cos110)x+28*sin110 y
B=63*cos(-30)x+63*sin(-30)
R=A+B=
you add them, like components to like components.
Polar: R@angle
R= sqsrt(xcomponents^2 + ycomponents^2)
angle=arctan(Y/x)
A = 28*Cos110 + 28*sin110 = -9.58 + 26.3i.
B = 63*Cos(-30) + 63*sin(-30) = 54.6 - 31.5i.
A+B = -9.58+20.3i + 54.6-31.6i = 45.0 - 11.3i.
Cartesian = (x,y) = (45,-11.3).
Polar: Tan A = Y/X,
A = ?.
R = 45/Cos A.
Polar = (R,A degrees).
Note: A is negative and in the 4th Quadrant.
To determine the sum of two vectors A and B, we need to add their corresponding components. In this case, we have:
Vector A = (28, 110°)
Vector B = (63, -30°)
To add the vectors, we can break them down into their Cartesian coordinates using the following formulas:
A = (Acosθ, Asinθ)
B = (Bcosφ, Bsinφ)
where A and B are the magnitudes of the vectors, and θ and φ are the angles associated with the vectors.
For vector A:
Ax = Acosθ = 28cos(110°) ≈ -3.22
Ay = Asinθ = 28sin(110°) ≈ 26.15
For vector B:
Bx = Bcosφ = 63cos(-30°) ≈ 54.6
By = Bsinφ = 63sin(-30°) ≈ -31.5
Now we can add the corresponding components:
Rx = Ax + Bx = -3.22 + 54.6 ≈ 51.38
Ry = Ay + By = 26.15 + (-31.5) ≈ -5.35
Therefore, the Cartesian representation of the sum A + B is (51.38, -5.35).
To find the polar representation, we can use the following formulas:
R = √(Rx^2 + Ry^2)
θ = tan^(-1)(Ry / Rx)
Using these formulas, we can calculate:
R = √((51.38)^2 + (-5.35)^2) ≈ 51.59
θ = tan^(-1)(-5.35 / 51.38) ≈ -5.97°
Note: The angle can be negative because the vector is pointing downwards.
Therefore, the polar representation of the sum A + B is approximately (51.59, -5.97°).