3.3 Vector Addition and Subtraction: Analytical Methods

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.