Vector A has a magnitude of 18 (in some unspecified units) and makes an angle of 25 with the x axis, and a vector B has a length of 24 and makes an angle of 70 with the x axis. Find the components of the vector C in the following:
(a) C = A + B
Cx =
Cy =
(b) C = A - B
Cx=
Cy=
The x component of A is 18 cos25
The y component of A is 18 sin25
The x component of B is 24 cos70
The y component of B is 24 sin70
Compute the numbers using trig functions and then compute C by adding or subtracting components.
To find the components of vector C, we need to break down each vector into its x and y components and then perform vector addition and subtraction.
(a) C = A + B:
To find the x and y components of vector A:
Ax = A * cos(theta), Ay = A * sin(theta)
where A is the magnitude of the vector A and theta is the angle it makes with the x-axis.
Plugging in the values: A = 18, theta = 25 degrees
Ax = 18 * cos(25) ≈ 15.92
Ay = 18 * sin(25) ≈ 7.87
To find the x and y components of vector B:
Bx = B * cos(theta), By = B * sin(theta)
where B is the magnitude of vector B and theta is the angle it makes with the x-axis.
Plugging in the values: B = 24, theta = 70 degrees
Bx = 24 * cos(70) ≈ 7.71
By = 24 * sin(70) ≈ 22.02
Now, let's add the x and y components of A and B to find the x and y components of vector C:
Cx = Ax + Bx ≈ 15.92 + 7.71 ≈ 23.63
Cy = Ay + By ≈ 7.87 + 22.02 ≈ 29.89
Therefore, the components of vector C in (a) are:
Cx ≈ 23.63
Cy ≈ 29.89
(b) C = A - B:
The process is the same as in (a), except we subtract the x and y components of B from the x and y components of A.
Cx = Ax - Bx ≈ 15.92 - 7.71 ≈ 8.21
Cy = Ay - By ≈ 7.87 - 22.02 ≈ -14.15
Therefore, the components of vector C in (b) are:
Cx ≈ 8.21
Cy ≈ -14.15