find the direction and magnitude of the vectors : a. A=(25m)x+(-12)y b. B=(2.0)x +(15)y c. A+B
To find the direction and magnitude of a vector, we can use the following formulas:
Magnitude of a vector A: |A| = sqrt(A_x^2 + A_y^2)
Direction of a vector A: θ = atan(A_y / A_x)
Let's apply these formulas to each vector:
a. A = (25m)x + (-12m)y
Magnitude of vector A:
|A| = sqrt((25m)^2 + (-12m)^2)
|A| = sqrt(625m^2 + 144m^2)
|A| = sqrt(769m^2)
|A| ≈ 27.73m
Direction of vector A:
θ = atan((-12m) / (25m))
θ ≈ -25.9° (in the fourth quadrant)
So, the magnitude of vector A is approximately 27.73m, and its direction is approximately -25.9°.
b. B = (2.0m)x + (15m)y
Magnitude of vector B:
|B| = sqrt((2.0m)^2 + (15m)^2)
|B| = sqrt(4.0m^2 + 225m^2)
|B| = sqrt(229m^2)
|B| ≈ 15.13m
Direction of vector B:
θ = atan((15m) / (2.0m))
θ ≈ 82.87° (in the first quadrant)
So, the magnitude of vector B is approximately 15.13m, and its direction is approximately 82.87°.
c. A + B = (25m)x + (-12m)y + (2.0m)x + (15m)y
Combining like terms, we get:
A + B = (25m + 2.0m)x + (-12m + 15m)y
A + B = (27m)x + (3m)y
Magnitude of vector A + B:
|A + B| = sqrt((27m)^2 + (3m)^2)
|A + B| = sqrt(729m^2 + 9m^2)
|A + B| = sqrt(738m^2)
|A + B| ≈ 27.14m
Direction of vector A + B:
θ = atan((3m) / (27m))
θ ≈ 6.43° (in the first quadrant)
So, the magnitude of vector A + B is approximately 27.14m, and its direction is approximately 6.43°.