Calculus II/Vector operations

Vector and its opposite vector edit

If there is a vector   then its opposite vector is   . The opposite vector has the same magnitude as vector's magnitude but with opposite in direction . Opposite vector of vector B is  

Sum and difference of 2 vectors edit


Dot product of two vectors edit

Definitioɲ edit

The scalar product or inner product or dot product of two vectors is defined as


where ː  is the angle between the two vectors (see Figure 2(b)).

If   and   are perpendicular to each other,   and  . Therefore,  .

The dot product therefore has the geometric interpretation as the length of the projection of   onto the unit vector   when the two vectors are placed so that they start from the same point.

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as


If the vector is   dimensional, the dot product is written as


Using the Einstein summation convention, we can also write the scalar product as


Dot product identities edit

  1.   (commutative law).
  2.   (distributive law).

Cross product of two vectors edit

Definition edit


The vector product (or cross product) of two vectors   and   is another vector   defined as


where ː  is the angle between   and  , and   is a unit vector perpendicular to the plane containing   and   in the right-handed sense (see Figure 3 for a geometric interpretation)

In terms of the orthonormal basis  , the cross product can be written in the form of a determinant


In index notation, the cross product can be written as


where   is the Levi-Civita symbol (also called the permutation symbol, alternating tensor). This latter expression is easy to remember if you recognize that xyz, yzx, and zxy are "positive" and the others are negative: xzy, yxz, zyx.

If  , then


Indentities edit


The rest of this resource has been moved to Vector calculus.