Determine the magnitude of the vector sum V=V1+V2 and the angle theta(x) which V makes counterclockwise from the positive x-axis.
V1 = 34 units and makes a 20 degree angle with the +x axis
V2 = 28 units and forms a 3,4,5 triangle with the -x axis
See previous post: Tue,8-26-14,10:29 PM.
To determine the magnitude of the vector sum V = V1 + V2 and the angle theta(x) that V makes counterclockwise from the positive x-axis, we can use vector addition and trigonometry.
1. Find the x-component and y-component of each vector:
- V1: Given that V1 has a magnitude of 34 units and makes a 20 degree angle with the +x axis, we can use trigonometry to find its x-component and y-component.
- x-component of V1 = magnitude of V1 * cos(angle) = 34 * cos(20°)
- y-component of V1 = magnitude of V1 * sin(angle) = 34 * sin(20°)
- V2: Given that V2 has a magnitude of 28 units and forms a 3, 4, 5 triangle with the -x axis, we can use the triangle's proportions to find its x-component and y-component.
- Since the triangle is 3, 4, 5, the x-component of V2 is -4/5 times the magnitude of V2 (-4/5 * 28)
- The y-component of V2 is -3/5 times the magnitude of V2 (-3/5 * 28)
2. Add the x-components and y-components of both vectors:
- Vx = x-component of V1 + x-component of V2
- Vy = y-component of V1 + y-component of V2
3. Calculate the magnitude of the vector sum V:
- Magnitude of V = sqrt(Vx^2 + Vy^2)
4. Calculate the angle theta(x) that V makes counterclockwise from the positive x-axis:
- Angle theta(x) = atan(Vy/Vx) (Inverse tangent of Vy/Vx)
Now let's perform the calculations:
V1:
- x-component of V1 = 34 * cos(20°)
- y-component of V1 = 34 * sin(20°)
V2:
- x-component of V2 = -4/5 * 28
- y-component of V2 = -3/5 * 28
Adding components:
- Vx = x-component of V1 + x-component of V2
- Vy = y-component of V1 + y-component of V2
Magnitude of V:
- Magnitude of V = sqrt(Vx^2 + Vy^2)
Angle theta(x):
- Theta(x) = atan(Vy/Vx)
Plug in the values and calculate the result.