Vector Addition And Trigonometry

Vectors can be described as directed segments and as ordered pairs.  However, since it is easier to add vectors as ordered pairs, you would want to translate the vectors from directed segments to ordered pairs. Translation from directed segments to ordered pairs can be done by using trigonometry.

* In a right triangle ABC with right angle C,
the tangent of angle A, written tan(A), is leg opposite angle A divided by (/) leg adjacent to angle A;
the sine of angle A, written sin(A), is leg opposite angle A / hypotenuse;
the cosine of angle A, written cos(A), is leg adjacent to angle A / hypotenuse.

Here is an example:
graph--sorry, Lynx users, math is graphical
Describe the vector OD above with an ordered pair.

The desired ordered pair contains the coordinates of point D. Let D = (x,y). From the graph you can see that x is positive and y is negative. The absolute values of x and y are OC and CD. Now use trigonometry.

OC/OD = cos(30°)
OC = OD*cos(30°)
OC = 25*cos(30°)

CD/OD = sin(30°)
CD = OD*sin(30°)
CD = 25*sin(30°)

A = (25*cos(30°), -25*sin(30°) = about (21.65, -12.5)
 

Great! Let me try some problems myself!


1. Find the components of the vector whose magnitude is 150 and whose direction is 10 degrees south of west.
About  (,)

2. Find the direction and magnitude of the vector (3,5) magnitude:    direction:  degrees north of east

3. To move a heavy box, one person pushes north with a force of 125 lb. A second person pushes northwest with a force a 100 lb. What is the magnitude and direction of the combined force?
The combined force is about  lbs. in a direction  degrees north of west.

4. From the top of a building, a person throws a ball horizontally at 88 feet per second. The force of gravity pulls it down 32 feet in the first second. After one second, how far and in what direction from the person will the ball be? about  feet away; about  degrees below horizontal

5. Find the components of the vector with magnitude 10 and the direction is 20 degrees north of west.  about (, )

Stumped? Take a look at the answer key!

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