A vector A has magnitude 5.0m and is directed east. Vector B has magnitude 4.0m and is directed 35° west of north.what is (a) magnitude.(b) the direction of a+b . (c) magnitude and (d) the direction of b-a
(a) To find the magnitude of vector A + B (a + b), we first need to break down vector B into its x and y components. The x-component of B can be found using the cosine function:
cos(35°) = adjacent / hypotenuse
cos(35°) = x-component / 4.0 m
Rearranging the equation, we get:
x-component = cos(35°) * 4.0 m
Calculating the x-component of B:
x-component = cos(35°) * 4.0 m ≈ 3.279 m
The y-component of B can be found using the sine function:
sin(35°) = opposite / hypotenuse
sin(35°) = y-component / 4.0 m
Rearranging the equation, we get:
y-component = sin(35°) * 4.0 m
Calculating the y-component of B:
y-component = sin(35°) * 4.0 m ≈ 2.284 m
Now, adding the x-components and y-components of A and B, respectively, to find the x and y components of A + B:
x-component of A + B = x-component of A + x-component of B
x-component of A + B = 5.0 m + 3.279 m ≈ 8.279 m
y-component of A + B = y-component of A + y-component of B
y-component of A + B = 0 m + 2.284 m ≈ 2.284 m
Using the Pythagorean theorem, we can find the magnitude of A + B:
magnitude of A + B = √(x-component of A + B)^2 + (y-component of A + B)^2
magnitude of A + B = √(8.279 m)^2 + (2.284 m)^2
magnitude of A + B ≈ √68.458 + 5.222 ≈ √73.679 ≈ 8.58 m
The magnitude of A + B is approximately 8.58 m.
(b) To find the direction of A + B, we need to calculate the angle it makes with the positive x-axis:
angle = arctan(y-component of A + B / x-component of A + B)
angle = arctan(2.284 m / 8.279 m)
angle ≈ 15.761°
The direction of A + B is approximately 15.761° east of north.
(c) To find the magnitude of B - A (b - a), we simply subtract the magnitude of A from the magnitude of B:
magnitude of B - A = magnitude of B - magnitude of A
magnitude of B - A = 4.0 m - 5.0 m
magnitude of B - A = -1.0 m
The magnitude of B - A is -1.0 m.
(d) To find the direction of B - A, we need to calculate the angle it makes with the positive x-axis. Since the magnitude is negative, the direction will be opposite to the direction of A + B. Therefore, the direction of B - A is approximately 180° - 15.761° = 164.239° west of north.