A vector, as opposed to a scalar, has a direction.. For example, velocity is a vector while speed is a scalar. When one says a car drives “at 60 mi/hr,” he/she is describing its “speed.” On the other hand, “60 mi/hr eastbound” is “velocity.”

Vector Graph 1





Vector Graph 2




or


The addition can be easily understood by breaking down each vector into components which are “projections” of the vector onto the x-y coordinates.

we can see that
and the direction of C points to .

The length of the vector is .


Product of Vectors
  1. Dot Product
    • Let A and B be 3-dimensional vectors represented in a rectangular coordinate system as:
      • ,
      • .
    • When i, j, k are unit vectors in the x, y, z directions respectively.


    • The dot product is defined as
      • .

    • Properties of the dot product
      • , where c is a scalar.

    • It can be proven that
      • , where θ is the angle betweeen A and B.
      • and
      • are the lengths of the respective vectors.


      It is obvious that the dot product of 2 orthogonal vectors equals 0.

  2. Cross Product

    • The cross product can be written as:

    • The value of the 3rd-order determinant is calculated:
      • ,

    • where the 2nd-order determinants are calculated:

    • It can be proven that
      • , where θ is the angle between the z vectors A and B.


    • The direction of the cross product can be memorized using the “rule of right thumb”:

    • It is obvious from the rule of right thumb that A × B points to the opposite direction of B × A. Since = 0 when θ = 0, cross product of 2 parallel vectors is 0.

    • Properties of Cross Products: