The most basic way to think about a vector is as a length (magnitude) associated with some direction.
There are more technical definitions, however, for the 21x series of courses, the above will suffice.
    Basis Vectors:
          The vectors i,j,k (or (1,0,0),(0,1,0),(0,0,1)) are called Basis Vectors. Collectively, they are called a Basis. All other vectors in a coordinate system are expressed in terms of these vectors. This basis has the following nice propertes:

• they have magnitude (length) 1. This means that when they are used in multiplication, they don't affect the physics.
• they are orthogonal (perpendicular). This means that when you travel along the x direction, you don't also move along the y direction.
• this basis does not change with position. They always point in the same direction, with the same magnitude.

    Notation:
        There are a few standard sets of notation for writing own vectors explicitly (in component form). Thesare basically         interchangeable. Vectors are usually written with boldface type, and I shal adhere to that standard.
        Lets' say we have a vector V = (V_x,V_y,V_z). We can also write these in the following ways:
        V = V_xi + V_yj + V_zk = V_xx + V_yy + V_zz = V₁e₁ + V₂e₂ + V₃e₃.
        The last notation, using "e"'s as the basis vectors is more general than cartesian geometries. They are used for any (3-dimensional) coordinate system. This is not to say that the vectors e are the same in any coordinate system. The notation is the same.

    Vector Algebra
    Now that we've defined the basics of vectors, it would be useful to know how to use them.
        Vector Addition:
            Let A = (A_x,A_y,A_z), and B = (B_x,B_y,B_z) be vectors in some cartesian vector space.
            Then C = A + B = (A_x + B_x, A_y + B_y, A_z + B_z).
You can visualize the addition of vectors by the following picture, which illustrates the so-called "parallelogram rule" for adding vectors.

    Figure 1.: Parallelogram Rule.
Slide B along A so that the tail (the end with the dot) of B coincides with the tip (end with the arrow) of A. This is why we say that vectors add "tip to tail".
        Vector Subtraction:
    Vector Subraction works in the way you'd expect. You take one vector, and multiply it by -1. Then add the two together. In symbols,
           D = A - B = (A_x - B_x, A_y - B_y, A_z - B_z).
A picture might help:

    Figure 2.: Two Ways to Visualize Vector Subtraction.
Start the tail of D at the tip of A, and arrange the tip of D to coincide with the tip of B. Or, you can multiply B by -1, and add it to A. Multiplication by a scalar is detailed below (as well as multiplication by a vector, for that matter).
        Vector Multiplication:
    There are three types of multiplication involving vectors.
              Multiplication by a Scalar
                    Let A be a vector, and "a" be a scalar. Then:
                        aA = (aA_x, aA_y, aA_z).
                    If a > 0, then the vector A does not change direction. It points along A, but with a different magnitude.
                    If a < 0, then the vector A again changes magnitude, but now it has rotated 180 degrees.
                    In either case, aA and A are parallel to each other.
              Multiplication by another vector:
    There are two ways to multiply one vector by another.
                   Dot Product (also called the Scalar Product):
    The scalar product is a measure of how much one vector lies along another. The defining property is that the Scalar Product of two vectors is a Scalar. It is not a vector. You can think of the scalar product as the projection of one vector along another.
                      A(dot)B = (A_xB_x, A_yB_y, A_zB_z) = ABcos(theta).
    Here I have written A for the magnitude of A, and B for the magnitude of B.
    The magnitude of a vector is defined by the following:
                        A = sqrt(A(dot)A).
    The magnitude is just the length of any vector. This is an example of how the dot product is used.
    For those of you continuing studies in physics, this will show up in a more generalized form, and be called an "inner product".
                  Cross Product (also called the Vector Product):
    The cross product is the vector multiplication rule that most students have the most difficulty with. One can construct, from any two vectors A and B, a parallelogram. The the cross product of these two vectors is another vector. The new vector has a magnitude which is equal to the area of the parallelogram. You can find the cross product the following way:
                               | i     j    k |
C = A(cross)B= det|A_x A_y A_z |
                               |B_x B_y B_z  |
By taking the determinant of this matrix, one obtains the Cross product. There is a simpler formula for the magnitude of C:
                        C = ABsin(theta).
    The more general term for the cross product is "outer product". Not that you need to know anything about this terminology for physics 21x.

    These are some mistakes that people commonly make when working with these multiplication rules:

• The Cross Product and Dot Product are only defined for multiplying two vectors together. You cannot cross a vector and a scalar. Similarly, you cannot dot a vector and a scalar. If you want to multiply a vector and a scalar, follow the first rule, above.
• Be very careful how you order the Cross Product. If you put the vectors involved in the Cross Product in the reverse order, you will end up with an extra minus sign. In other words, A(cross)B is not equal to B(cross)A.