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Chapter 15: Addition Methods Using the Components of Vectors

- A coordinate system can be used to resolve a single vector in space.
- We can review the methods at the beginning of the section.
- We can see that it is composed of an x and a y with a displacement of 100 meters northeast and a 45 degree angle with the positive x - axis.

- If we projected a line down from the head of the given vector to the x- axis, we would be able to construct the two parallel components.

The magnitude R may be written as R-

- When adding two or more vectors, this method can be useful.
- The result of the addition and subtraction of the respective components will be found if we are given two vectors.

- An example of this method is shown.
- A- and B- represent tensions in two ropes applied at the origin of the coordinate system.
- The force between these tensions can be found by determining the x and y components of the given vectors.

- The angle with the positive x - axis is about 105 degrees.

- There are quantities that have both magnitude and direction.

- There are components that represent the magnitude and direction of aVector along an axis.

- There are quantities without direction.

- Force, displacement, and velocity are examples.

- The examples of scalars are distance, speed, and mass.

- The tip-to-tail method can be used to add Vectors.

- The resulting two vectors are the same as the one obtained by "adding" them.

- The sum of the magnitudes of the two vectors is their resultant.

- The result is equal to the difference of the magnitudes of the two vectors.

- A mass sliding down an inclined plane is an example of a rotating system with the " x - axis" parallel to the incline.

- Pick the right scale for your diagram.
- A scale of 1 cm is appropriate for a displacement problem.

- As they correspond to geographical directions, recognize the given orientation of the vectors.
- Keep the same directions as you draw your triangle.

- The result from the tail of the first vector to the head of the last one should be connected.

- Determine the components of the vectors.
- In most cases, components can be used to simplify problems.

- The components of a vector are Ay and Ax.

- The angle of 20deg is the magnitude of 17 units and the positive x - axis.

- There is a magnitude of 10 units and an angle of 30 degrees with the horizontal x - axis.
- The angle of 50deg with the negative x - axis is achieved by a magnitude of 25 units.

- There are two concurrent units with magnitudes of 3 and 8 units.
- The difference is 8 units.

- Three forces act on the same point.

- First base is close to home plate on a baseball field.
- A batter runs to first base after a hit.
- She runs past the base and back to stand on it.

- The magnitude and direction of the two concurrent forces can be found using the algebraic method of components.

- The figure is not drawn to scale.

- A- and B- are attached at the tails with an angle between them.

- 3 and th are rounded off to 72 degrees.

- The magnitude is equal to the numerical difference between the magnitudes.
- If the angle were specified, the maximum and minimum set could include the given value of 35.
- 35 in their range is not included in the maximum and resultant minimums of the others.

- When the known values are used, they are values of 17 cos 20.

We have a number of cos and a number of sin and Bx and Ay and Ay and Bx and By and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay and Ay

- The result is Cx + Bx + 8.66 - 16.06, so Cx is 7.4 units and Cy is 5 units.

- The cos are 0.1875 and 79deg.

- The only requirement is that the vectors are the same size as the shuffled ones.
- I, III, and IV are the same displacements that are listed in different orders.
- Three sets will produce the same result.

- The result is drawn from the tail of the first to the head of the second.
- The first choice is the horizontal one, the second is the general direction of the result.

- The result should be drawn from the tail of the first to the head of the last.

- 10 N force and 20 N force are used.

- The components are given by the equations.

- The situation is shown.
- A-B- is the third side of the triangle.

- The components of A--B- are Ax and Ay.

- The parallelogram method of construction can be used to add any two vectors.
- A plane figure is a parallelogram.
- The diagonal of this parallelogram shows the result of two vectors.
- To add up to zero, the third remaining vectors must be equal in magnitude but opposite in direction.
- To achieve this result, all three must lie in the same plane.

- They can be in any orientation.
- No coordinate system is required.

- To form these components, a coordinate system must be specified.

- The square root of the sum of the squares of the magnitudes of its components is the magnitude of a vector.
- If one of the components is nonzero, the magnitude of the whole vector must be nonzero as well.

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