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=
(a) θ=25° φ=70°
Cx = Ax + Bx = |A|cos(θ) + |B|cos(φ)
=18cos(25)+24cos(70)
= 16.31 + 8.21
= 24.52
Cy = Ay + By = |A|sin(θ) + |B|sin(φ)
= 18sin(25)+24sin(70)
= 7.61 + 22.55
= 30.16
(b) similar to (a), but subtract instead of add.
To find the components of vector C in both cases, we need to break down the vectors A and B into their x and y components, and then perform the corresponding addition or subtraction operation.
Let's start by finding the x and y components of vector A:
Magnitude of vector A = 18
To find the x component of vector A, we use the cosine of the angle it makes with the x axis:
Ax = magnitude of A * cos(angle)
= 18 * cos(25)
To find the y component of vector A, we use the sine of the angle it makes with the x axis:
Ay = magnitude of A * sin(angle)
= 18 * sin(25)
Next, let's find the x and y components of vector B:
Magnitude of vector B = 24
To find the x component of vector B, we use the cosine of the angle it makes with the x axis:
Bx = magnitude of B * cos(angle)
= 24 * cos(70)
To find the y component of vector B, we use the sine of the angle it makes with the x axis:
By = magnitude of B * sin(angle)
= 24 * sin(70)
Now, let's calculate the components of vector C in both cases:
(a) C = A + B
To find the x component of vector C, we add the x components of A and B:
Cx = Ax + Bx
To find the y component of vector C, we add the y components of A and B:
Cy = Ay + By
(b) C = A - B
To find the x component of vector C, we subtract the x component of B from the x component of A:
Cx = Ax - Bx
To find the y component of vector C, we subtract the y component of B from the y component of A:
Cy = Ay - By
Now you can substitute the values we calculated earlier to find the x and y components of vector C in both cases.