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.

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.

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

Properties of Vector Addition Theorem:
(A) Vector addition is commutative.
(B) Vector addition is associative.
(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!

Stumped? Take a look at the answer key!