A diagram has vector A at 30 degrees above the positive x axis. length = 3 m
Vector B is ON the negative x-axis. length = 1 m
Vector C is ON the negative y-axis. Length is 2 m
What is the direction in degrees of vector a + vector B + vector C?
In x direction: 3 cos 30 - 1 = 1.6
In y direction: 3 sin 30 - 2 = 1.5-2 = -.5
tan^-1 (-.5/1.6) = - 17.35 degrees
or east 17.35 deg south
To find the direction of vector a + vector B + vector C, we need to add the individual directions of each vector.
Vector A is at 30 degrees above the positive x-axis.
Vector B is on the negative x-axis, which means it is directed towards the negative y-axis. Therefore, its direction is -90 degrees.
Vector C is on the negative y-axis, which means it is directed towards the negative x-axis. Therefore, its direction is 180 degrees.
To determine the direction of the sum, we add the angles:
30 degrees (vector A) + (-90 degrees) (vector B) + 180 degrees (vector C) = 30 - 90 + 180 = 120 degrees.
Therefore, the direction of vector a + vector B + vector C is 120 degrees.
To find the direction in degrees of the resultant vector A + B + C, we need to first find the magnitudes and directions of each vector, and then add them together.
1) Vector A:
- We are given the direction as 30 degrees above the positive x-axis.
- The length of vector A is given as 3 meters.
2) Vector B:
- We are told that vector B is on the negative x-axis.
- The length of vector B is given as 1 meter.
3) Vector C:
- We are informed that vector C is on the negative y-axis.
- The length of vector C is given as 2 meters.
Now, let's find the magnitudes and directions of each vector:
1) For vector A:
- The magnitude of vector A is given as 3 meters.
- The direction is 30 degrees above the positive x-axis.
2) For vector B:
- The magnitude of vector B is given as 1 meter.
- Since vector B is on the negative x-axis, its direction is 180 degrees.
3) For vector C:
- The magnitude of vector C is given as 2 meters.
- Since vector C is on the negative y-axis, its direction is 270 degrees.
Now, let's add these vectors together to find the resultant vector:
A + B + C = (3 m @ 30 degrees) + (1 m @ 180 degrees) + (2 m @ 270 degrees)
To add these vectors, we can break them down into horizontal and vertical components and add them separately. Then, we can find the magnitude and direction of the resultant vector.
Horizontal component:
- A: 3 cos(30 degrees) = 3 * √(3) / 2 = 3√(3) / 2
- B: 1 cos(180 degrees) = -1
- C: 0
Vertical component:
- A: 3 sin(30 degrees) = 3 * 1 / 2 = 3 / 2
- B: 1 sin(180 degrees) = 0
- C: 2
Adding the horizontal components:
3√(3) / 2 - 1 + 0 = 3√(3) / 2 - 1
Adding the vertical components:
3 / 2 + 0 + 2 = 3 / 2 + 2 = 7 / 2
Now, let's find the magnitude and direction of the resultant vector using the Pythagorean theorem and inverse tangent:
Magnitude:
√[(3√(3) / 2 - 1)^2 + (7 / 2)^2] = √[(9/4) - (3√(3) / 2) + 1 + 49/4] = √(59/4 - 3√(3) / 2)
Direction:
arctan[(3 / 2) / (3√(3) / 2 - 1)]
Simplifying the direction:
arctan[(3 / 2) / (3√(3) / 2 - 1)] = arctan[(3 / 2) / (3√(3) / 2 - 1)] * (180 / π)
Now, you can use a calculator to find the approximate value of the magnitude and direction.