Inner Products and Orthogonality

The scalar product (or dot product or inner product ) of two row vectors (or two column vectors) is defined as the sum of aibi. In order to take the scalar product of two vectors, they must have the same size. Note also that the scalar product is a number, not a vector. The standard inner product on R^n is precisely the dot product of vectors that you may remenber from calculus:

In[56]:=

  v={v1, v2, v3}
  w={w1, w2, w3}

Out[56]=

  {v1, v2, v3}

Out[57]=

  {w1, w2, w3}

In[58]:=

v.w

Out[58]=

  v1 w1 + v2 w2 + v3 w3

Two vectors are said to be orthogonal if their innerproduct is zero. This is a generalization of perpendicular lines. Orthogonal vectors play an important role in theory and applications. Two subspaces are orthogonal if all the vectors in one are orthogonal to all the vectors in the other.

The vectors {3,2} and {-2,3} are orthogonal.

In[59]:=

  p=ListPlot[{{3,2},{-2,3}},DisplayFunction->Identity, 
    PlotStyle->{RGBColor[1,0,0],PointSize[.02]}];
  q=ParametricPlot[{t,2 t/3},{t,0, 3},DisplayFunction->Identity];
  r=ParametricPlot[{t,-3 t/2},{t,-2, 0},DisplayFunction->Identity];
  Show[p,q,r,DisplayFunction->$DisplayFunction, 
  AspectRatio->Automatic]

Out[60]=

  -Graphics-

Up to Fundamental Theorem of Linear Algebra