Vectors
A vector is a quantity that has both magnitude and direction. A scalar, for comparison, has only magnitude. Velocity and displacement are everyday examples of vectors. An arrow with a direction is used to draw a vector. An arrow above a letter is used as the notation for a vector. In publications, a bold letter like r is often used instead because it is easier to type.
There are two kinds of vectors. One is called a polar vector; the other is called an axial vector or pseudo-vector. They both obey vector algebra, but they have different behaviors in a symmetric transformation called space inversion. Velocity and displacement are both vectors, while angular velocity, scalars, and infinitesimal angular displacement are examples of pseudo-vectors.
Vectors also have algebraic operations. The ones used most widely are addition, subtraction, and multiplication (dot product and cross product). The set of rules for these operations is called vector algebra. Division of vectors is not defined. There are two methods to do these operations on vectors, geometric method and matrix method. The matrix method is used most frequently in physics, but the geometric method is much more intuitive. The operations below are illustrated using the geometric method.
Vectors can be added up in the following way. First draw out the vectors as arrows one by one, putting the tail of the next arrow right at the head of the previous one. After this is done, draw an arrow from the tail of the first vector to the head of the last one. That final arrow is the sum of all the vectors. The order of vectors does not matter. For example, in the addition of velocities, one can suggest that someone walking for three miles north followed by four miles east will end up at the same point as somebody walking five miles in the appropriate north-easterly direction.
A vector can also be subtracted from another vector. To subtract one vector from another, reverse the first vector's direction and subtract it from the other. For example, to calculate a - b, draw the two vectors as arrows, making their tails starting at the same point, then draw an arrow from the head of b to the head of a (order is important here). That final arrow is a - b. If the order of drawing at the final step is reversed, the result will be b - a instead.
Vector products come in two types. Dot products, also called inner products, give a scalar (an ordinary quality) as a result. For example, 12 km is a scalar product, while 12 km northwest is a vector. One can multiply the scalar of two vectors using the dot product function. Cross product functions, on the other hand, give a third vector as a result.
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