Vectors - Fundamentals and Operations

A vector quantity is a quantity which is fully described by both magnitude and direction.

Vector quantities are often represented by scaled vector diagrams.

Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction.

There are several characteristics of this diagram which make it an appropriately drawn vector diagram.

A. A scale is clearly listed

B. An arrow (with arrowhead) is drawn in a specified direction; thus,

the vector has a head and a tail. C. The magnitude and direction of the vector is clearly labeled; in this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

Vector Components

A vector is a quantity which has magnitude and direction. Displacement, velocity, acceleration, and force are the vector quantities which we will discuss in our course.

Resultants

The resultant is the vector sum of two or more vectors. It is the result of adding vectors together.

Conventions for Describing Directions of Vectors

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle)

An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors:

20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m

The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram.

SCALE: 1 cm = 5 m

Interestingly enough, the order in which three vectors are added is insignificant; the resultant will still have the same magnitude and direction. For example, consider the addition of the same three vectors in a different order.

15 m, 210 deg. + 25 m, 300 deg. + 20 m, 45 deg.SCALE: 1 cm = 5 m

When added together in this different order, these same three vectors still produce a resultant with the same magnitude and direction as before (22 m, 310 deg.). The order in which vectors are added using the head-to-tail method is insignificant.

SCALE: 1 cm = 5 m

Example

The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale

The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector.

trigonometric functions relate the length of the sides of a right triangle to the angle of a right triangle. As such, trigonometric functions can be used to determine the length of the sides of a right triangle if one angle and the length of one side are known.

The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors which make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors which are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation which relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

Oblique Triangles

An oblique triangle is any triangle that is not a right triangle. It could be an acute triangle (all threee angles of the triangle are less than right angles) or it could be an obtuse triangle (one of the three angles is greater than a right angle).

Solving oblique triangles

The trigonometry of oblique triangles is not as simple of that of right triangles, but there are two theorems of geometry that give useful laws of trigonometry. These are called the "law of cosines" and the "law of sines."

The law of cosinesThis is a simply stated equation:

c2 = a2 + b2 – 2ab cos C

It looks like the Pythagorean theorem except for the last term, and if C happens to be a right angle, that last term disappears (since the cosine of 90° is 0), so the law of cosines is actually a generalization of the Pythagorean theorem.

Note that each triangle gives three equations for the law of cosines since you can permute the letters as you like.

The other two versions are then a2 = b2 + c2 – 2bc cos A, and b2 = c2 + a2 – 2ca cos B.

The law of sines

The law of sines is also a simply stated equation

sin a/a = sin b/b = sin c/c

You can use the law of sines in two ways

First, if you know two angles and the side opposite one of them, then you can determine the side opposite the other one of them. For instance, if angle A = 30°, angle B = 45°, and side a = 16, then the law of sines says (sin 30°)/16 = (sin 45°)/b. Solving for b gives b = 16(sin 45°)/(sin 30°) = 22.6274.

The Plane and The Wind

Problems

http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l1a.html

http://aleph0.clarku.edu/~djoyce/java/trig/oblique.html