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Chapter 12: Vectors

- A directed line segment is a way of locating the position of an object.
- If you draw an arrow, starting from the origin to a point in space, you have defined a position vector R-, whose magnitude is given by R- and is equal to the linear distance between the origin and the point.
- The angle that the arrow makes with the positive x - axis is identified by the Greek letter th.
- A quantity with both magnitude and direction is called a vector quantity.
- Any quantity with a single magnitude is a scalar quantity.
- There are examples of force, velocity, weight, and displacement.
- Mass, distance, speed, and energy are examples of scalar quantities.

- There are both magnitude and direction.
- There is only magnitude for scurrs.

- The polar coordinate system is an alternative coordinate system if we designate the magnitude of the R- as r and the direction angle as th.
- To find the point in the system in two dimensions, you have to move x units horizontally and y units vertically.

- You need to be able to use the math in this section to go back and forth between the two representations.

- The ability to combine is important in physics.
- The addition of two numbers is not the same as the addition of two vectors.
- We must be careful to preserve the directions of the vectors relative to our chosen frame of reference when we state that we want to form the third vector C-.

- From head to tail, they add something constructive.
- The sum is called the resultant.

- The construction of a vector diagram is one way to do this "addition".
- We will look at a number of examples.

- A girl is walking from her house to an additional two blocks east.
- We need to choose a scale to represent the magnitude lengths of the vectors.
- To be equal to one block of distance, we need to call the scale "one unit" or just "one unit".
- "East" is to the right and "North" is to the left.
- There is a diagram for this set of displacements.
- The resultant has both magnitude and direction.

- Suppose the girl walks five blocks east and then two blocks northeast, 45 degrees north of east.
- The diagram will look like that.
- Some texts use the "parallelogram" method of construction, in which the result is drawn from the tail of the first to the head of the second.
- The result is measured from the horizontal axis.

- Suppose the girl walks five blocks east and two blocks north.

- Suppose the girl walks five blocks east and two blocks west.
- The starting point will be three blocks east of her final displacement.

- The two magnitudes 5 and 2 could add to any value between 7 and 3, depending on their relative orientation.

- The magnitude of their decreases are deduced from these examples.
- The magnitude of the result between the two vectors is a maximum and a minimum when they are both in the same direction and at a relative angle of 180 degrees.
- It is important to remember that the system must have a scale that is suitable for it.

- In the above examples, the result was constructed using a diagram.
- The magnitude of the result was the length of the vector drawn from the tail of the first to the head of the second.
- If we sketch such a situation, we can form a triangle that is related to the law of cosines and the law of sines.

- A, B, and C are the angles of the triangle with arbitrary sides.

- The angle A is equal to 12.45 degrees north of east.

- Multiple vectors can be added in any order as long as they are constructed head to tail.
- The diagram will be a closed geometric figure if the result is zero.
- This can be seen in the case of three vectors forming a closed triangle.

- If the girl walks five blocks east, two blocks north, and five blocks west, the diagram will look like, with the displacement being equal to two blocks north.

- The addition of a negative vector is a subtraction.

- A negative vector is simply one of the same magnitude but directed in a different direction.
- The result of this operation is the same as in the example that follows since it has the same magnitude and direction.

- 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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