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3.3 Vector Addition and Subtraction: Analytical Methods

- The part of the graphical technique that is retained is the representation of the vectors by arrows.
- Analytical methods are more precise and accurate than graphical methods, which are limited by the accuracy with which a drawing can be made.
- The accuracy and precision with which physical quantities are known are the limits of analytical methods.

- Analytical techniques and right triangles are related to physics because the motions of the right triangles are independent.
- We often need to separate a component from a component.

- The right triangle is formed by these vectors.
- The analytical relationships are summarized.

- The relationship between the components and the result is only for the quantities of magnitude and direction.
- The magnitudes alone do not apply to the relationship.
- If east, north, and north-eastern, then it is true.
- The sum of the magnitudes is not equal.

- The magnitude and angle of the vector are known if it is known.

- The angle with trigonometric identities can be related to the magnitudes of the vector components.

- The total displacement of the person walking in a city is represented by the vector.

- In this example, we can use the relationships to determine the magnitude of the horizontal and vertical component vectors.

- If the components are known, then they can also be found.

- Once the horizontal and vertical components have been determined, the magnitude and direction can be determined.

- The Pythagorean Theorem relates the legs of a right triangle to the length of the hypotenuse.

- The direction is the same as before.

- Both processes are crucial to analytical methods.

- In Figure 3.30, you can see how to add the vectors using the components.

- The resulting or total displacement of the two legs of a walk is called a vecis.
- The magnitude and direction can be determined using analytical methods.

- The total displacement is the number of legs of a walk.

- There are many ways to arrive at the same point.
- The person could have walked first in the x-direction and then in the y-direction.

- When you use the analytical method, you can determine the magnitude and direction of a vector.

- The x- and y-axes will be used in the problem.

- Determine the horizontal and vertical components of the vectors before adding them.

- Adding the components of the individual vectors along the axis will find the components of the resultant.

- The magnitude of the vectors and add to give the magnitude in the horizontal direction.

- The x- axis has components along the same line that can be added to one another like ordinary numbers.
- The components along the y- axis are the same.
- It's easier to add components that are along common axes.
- The magnitude and direction can be found now that the components are known.

- The following example shows how to add vectors using components.

- The x- and y-axes are in the opposite directions.
- A person walks north of east in the first leg of a walk.
- The second leg is a displacement north of east.

- The magnitude and direction are north of the x- axis.
- The magnitude and direction can be determined using analytical methods.

- The x- and y-axes represent the way to get to the same ending point.

- They are combined to produce a result.

- We first find the components of the x- and y-axes.

- This example shows the addition of components.
- The addition of a negative Vector is very similar to the addition of a perpendicular component.

- Subtraction can be accomplished by the addition of a negative vector.
- The method for addition and subtraction is the same.
- The negatives of the components are the components.

- The method outlined above is the same as for addition.

- Punctual quantities are often independent of one another, which makes analyzing them useful in many areas of physics.
- The next module, Projectile Motion, is one of many in which the components are used to make the picture clear and simplify the physics.

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