VECTORS:

Beware the slings and vectors of outrageous forces. -almost Shakespeare

In the most general sense, vectors are things that act like baskets of fruit. When you add them, their components add separately: Basket A has 2 apples, 3 oranges and a banana. Basket B has 5 oranges and a pear. Then A+B is 2 apples, 8 oranges, a banana and a pear. This is like the state vector in quantum mechanics. (Incidentally, if IOU an apple, this could be considered a negative apple, so fruit baskets are not limited to positive components.) By the way, do not tell your physics instructor that vectors are anything like baskets of fruit. (S)he will have a fit!

Until you get to quantum mechanics, all vectors in physics have a geometrical interpretation- they have a direction in space and they have a maximum of three components, one for each of the three dimensions of space. Let's stick to 2 dimensions for now. The convention is to list the x (horizontal) component first, then y. Thus, the vector (4, -1) can be pictured as an arrow pointing downhill to the right: 4 units to the right and 1 down. We do not need to use horizontal and vertical; any perpendicular directions are OK. For inclined plane problems, for example, it is usually best to use parallel and perpendicular to the plane.

The magnitude of a vector is represented by the length of the arrow, and it is found by Pythagorean Theorem. The (4, -1) has a magnitude equal to the square root of 17, or (17)^1/2.

To really get into vectors in a big way, you need to know a little trigonometry. Draw a right triangle and choose one of the acute angles (acute means less than 90 degrees). Label the side across from your chosen angle be called the side opposite. The other two sides are the hypotenuse (the longest) and the side adjacent. The sine of the angle is the side opposite divided by the hypotenuse, the cosine is the side adjacent divided by the hypotenuse, and the tangent is the side opposite divided by the side adjacent. Some people memorize SOH CAH TOA as an aid for working with these things, but you need to really know them in your gut. Given one side and one acute angle of a right triangle, you should be able to calculate the other two sides efficiently.

That's all the trig you need. Now if a vector is 5.0 units long and 20 degrees above horizontal, then the horizontal component is 5cos20^o and vertical is 5sin20^o. Check it out. Sin20^o is 0.342 and cos20^o is 0.940. From this we get a horizontal component (x component) of 4.7 and y component of 1.7. Since these form the two legs of a right triangle, the sum of their squares should be the square of the hypotenuse, 25. Check this (small rounding-off error).

ADDING VECTORS: All you do is add corresponding components. (4, -1) + (-6, 5) = (-2, 4). Another thing you need to know is the geometrical picture of this. Take some graph paper. Choose a starting point. Draw (4, -1) by drawing an arrow from your starting point to the point 4 units to the right and 1 unit down. Then from that last point, draw an arrow (-6, 5) by going 6 units to the left and 5 up. The end of that vector will then be 2 units to the left of and 4 units above your original starting point. So if you draw the vector (-2,4) from your original point, it goes to the tip of the 2nd arrow. This geometrical method works with any number of vectors (often called the polygon method). The vector sum is called the resultant, often symbolized by R:

Here is the component method for adding two vectors, magnitudes A and B, with angle q between them. The component method works for any number of vectors.

Here is a method utilizing the law of cosines. Use it for adding two vectors, or don't bother with it. The component method is more efficient for adding 3 or more vectors.

Here is the polygon method or diagram method of vector addition.

SUBTRACTING VECTORS: Simply reverse the direction of the one subtracted and add. A - B is the same as A + (-B).
(-B) is identical to B except that it is in the opposite direction.

MULTIPLYING VECTORS: Let us put this off until the subject of work and energy comes up for the scalar product (or dot product) and the subject of torque for the vector product (or cross product). Don't click on these now.

DIVIDING VECTORS: I dunno.

THE VECTOR AS A SIMPLE TENSOR: U don't wanna know.

THE VECTOR IN LINEAR ALGEBRA (matrix stuff): Row vectors and column vectors. Don't get into that now.

THE VECTOR IN QUANTUM MECHANICS: To illustrate the principles I have a crap game (dice) analyzed using these things. It would help to know about matrices and things, so you probably don't want to go there.

Don't click on this or a vector will stab you in the back.

My main pages: mechanics
fluids, heat, electricity and magnetism
vibrations and waves
quantum
index and quickie definitions and explanations

Comments, questions: email fredrick.gram @tri-c.edu (remove space before @)