# Vectors

When a basketball player dribbles the ball at 5 mph going east, the player's route is a vector. If the player turns and dribbles the ball 5 mph southward, that is another vector. The resultant route of the player or the resultant of the vector can be explained with mathmatics.

A vector is a quantity that has both magnitude and direction. Here are some examples.

 Vector Name Vector Magnitude Vector Direction gravity on earth 9.8 meters per second down southeast wind at 40 miles per hour 40 mph SE a person pushing a box north with a force of 100 pounds 100 lbs north

Vectors are represented by directed line segments. A directed line segments is drawn like a normal line segment, except that it has an arrow at one endpoint.

For instance, drawn below is the directed line segment CA representing a northeast wind of 25 mph. Point C is the initial point of the vector and A is its terminal point. Its direction is 60 degrees north of east (or 30 degrees east of north) and its magnitude is 25, the distance CA.

* The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.
* The sum of two noncollinear vectors OA and OB is the vector OC such that OACB is a parallelogram.

Any vector can be placed so that its initial point is the origin. If its terminal point is
(a,b), the vector is named (a,b). The horizontal component of (a,b) is a; the vertical component is b. This is the ordered pair description of a vector. The vector (a,b) can be interpreted as the resultant of a horizontal force a and a vertical force b.

The sum of the vectors (a,b) and (c,d) is the vector (a + c, b + d).

(C) (0,0) is an identity for vector addition
(D) Every vector (a,b) has an additive inverse (-a,-b).

* Let k be a real number and (a,b) be a vector. Then k(a,b), the scalar multiple of k and (a,b), is the vector (ka,kb).

## Great! Let me try some problems myself!

1. Add the vectors (0,0) and (.5,12) (,)

2. A plane is traveling at a ground speed (horizontally) of 350 mph and is descending at 1000 feet per minute. Convert these quantities to the same unit to find the slope of the vector describing the plane's path. about

3. A motorboat is traveling up a river at a speed that would be 25 miles an hour in still water. There is a current of 15 miles per hour coming down the river. How far will the motorboat travel in an hour? miles

4. The same motorboat from question 5 is traveling down the river with the same speed with the same current. How far will the motorboat travel in an hour? miles

Stumped? Take a look at the answer key!

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