THREE DIFFERENT METHODS.
Components*
Tip-To-Toe*
Parallelogram*
Useful info.
Vector-
A quantity that has a magnitude and direction. Velocity, acceleration,
and force are a few examples. Other quantities such as time, mass, and
temperature have quantity and are called scalars. The graph of a vector
is shown as an arrow with the tip pointed in the direction of the quantity
and the length representing the magnitude.
Component Method- This Method is the most precise and mathematical method of the three. In this method, all of the x-components and all of the y-components are added up. To break a two dimensional vector into its components, the equations
cos q = x-component / hypotenuse
sin q = y-component / hypotenuse
and solve for the x and y components. Then do the same for the next vector and add the x-components from the two vectors and the y-components from the vectors. Then to find the angle, use the equation,
tan angle = y-component / x-component
and to find the magnitude, use the Pythagorean theorem.
Example: A
sailboat sails 4 miles from the pier at 20 degrees north of east. Then,
it sails 5 miles 35 degrees north of west. What is the boat’s displacement
now?
cos 20 = x/4
4 cos 20 = x x = 3.76
sin 20 = y/4
4 sin 20 = y y = 1.37
180-35 = 145
cos 145 = x/5
5 cos 145 = x x = -4.10
sin 145 = y/5
5 sin 145 = y y = 2.87
Add the first x with the second x and the first y with the second y.
X= 3.76 + -4.10 = -0.34
Y= 1.37 + 2.87 = 4.24
tan angle = 4.24/-0.34 angle = 94.6 degrees.
(-0.34)^2 + (4.24)^2 = d^2 distance = 4.25 miles
Tip-To-Toe Method- This Method is not extremely precise, since you just use a protractor and a ruler. You would first draw a graph, then decide your scale. For example, if you tried the earlier example using this method, you could say that one mile was represented on the graph by one inch. Start anywhere on the graph, preferably the origin. Find 20 degrees and draw a line 4 inches long with an arrow on the end. The tip of this vector will be the toe of the next vector. Start at the tip of the first vector and draw a five inch line at 145 degrees. You can the measure the angle and distance of the resulting
point from
the origin. It probably won’t be exact but it should be close.
Parallelogram Method- This next method is more of visual method and is probably the least mathematical of the three. The two vectors are drawn on a graph, both starting at a common point. A parallelogram is constructed using the two vectors as adjacent sides. The point opposite the origin is the endpoint of the sum vector.
Jason Millett
Giancoli Physics
Principles with Applications 5e
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