On graph paper, identify the point (3, 4).
Draw the segment from the origin to the point and put a little arrow at (3, 4). Label the vector "v".

Draw the segment from (0, 0) to (3, 0) and attach an arrow head; draw the segment from (3, 0) to (3, 4) and draw an arrow head. Those two vectors can be thought of as the components of the first vector. Also, the first vector can be regarded as the resultant of the other two vectors.

There are two fundamental processes involving vectors. One prcess is to take a given vector (magnitude and direction are given) and break it down into horizontal and vertical components. The components are also vectors.

The other process is to work the other way around. That is, take component vectors (horizontal and vertical vectors) and combine them into a single vector called the resultant vector.

Let's go through both processes using the little illustration above. First, though, we need some information. Using the Pythagorean Theorem, you can find that the distance from the origin to (3, 4) is 5. Then, using any trig ratio, you can find that the angle formed inside the right triangle at the origin is approximately 53.13 degrees.

So, the first vector, the one from the origin to (3, 4), has a direction of 53.13 degrees and a force of 5 and we said above that we are calling it vector "v".

Suppose we don't have the component vectors and we have to figure them out on our own from the first vector. Suppose we don't even have the coordinates of the terminal point, (3, 4) nor the legs of the right triangle. All we have is the vector diagram, the angle of 53.13 degrees, and the magnitude of 5.

The components can be found using the form,

v = < horizontal component, vertical component >

v = < force (cos θ), force (sin θ) >

or

v = force < cos θ, sin θ >

Using the calculator, the component form is

v = < 3, 4 >

[The values are not exact on the calculator because the angle of 53.13 is an approximation.]

Another notation associated with

v = < 3, 4 > is

v = 3i + 4j

The notations are interchangeable with one a little more easy to use than the other in certain situations.

Now let's change the situation. Suppose we begin with the components and we want to write the resultant.

The magnitude, ||v||, is found using the Pythagorean Theorem because the component are horizontal and vertical (the legs of a right triangle) and the resultant is the hypotenuse. So,

||v|| = sqrt [(horiz component)^{2} + (vert component)^{2}]

||v|| = sqrt (3^{2} + 4^{2})

||v|| = sqrt (9 + 16)

||v|| = sqrt 25

||v|| = 5

The direction of the resultant is found using the relationship

tan θ = opp / adj

tan θ = vertical component / horizontal component

tan θ = 4/3

θ = inverse tan (4/3)

θ = 53.13 degrees (approximately)

Next topic.

When you need to find the resultant of two vectors that are not horizontal and vertical, write the two vectors in component form and then add the components together.

That is, suppose the vectors are "u" and "v".

Suppose vector u has a force of 70 and a direction of 10 degrees and vector v has a force of 50 and a direction of 60 degrees.

Component forms:

u = < force (cos θ), force (sin θ) >

u = < 70 cos 10, 70 sin 10 >

u = < 68.94, 12.16 >

or

u = 68.94 i + 12.16 j

The resultant vector is

r = u + v

r = < 68.94, 12.16 > + < 25, 43.3 >

r = < (68.94 + 25), (12.16 + 43.3) >

r = < 93.94, 55.46 >

or

r = u + v

r = (68.94 i + 12.16 j) + (25 i + 43.3 j)

r = 93.94 i + 55.46 j

||r|| = sqrt (93.94^{2} + 55.46^{2})

||r|| = 109.09

tan θ = 55.46 / 93.94

θ = inverse tan (55.46 / 93.94)

θ = 30.56

So, the resultant vector is a force of 109.09 in a direction of 30.56 degrees.

I hope this clears up alot of confusion about vectors.