Scalar Product

(A.23) |

for general vectors and . Is invariant under transformation, as must be the case if it is a scalar number? Let us consider an example. Suppose that and . It is easily seen that . Let us now rotate the coordinate axes through about . In the new coordinate system, and , giving . Clearly, is not invariant under rotational transformation, so the previous definition is a bad one.

Consider, now,
the *dot product* or *scalar product*:

Let us rotate the coordinate axes though degrees about . According to Equations (A.20)-(A.22), takes the form

(A.25) |

in the new coordinate system. Thus, is invariant under rotation about . It is easily demonstrated that it is also invariant under rotation about and . We conclude that is a true scalar, and that the definition (A.24) is a good one. Incidentally, is the only simple combination of the components of two vectors that transforms like a scalar. It is readily shown that the dot product is commutative and distributive: that is,

(A.26) |

The associative property is meaningless for the dot product, because we cannot have , as is scalar.

We have shown that the dot product is coordinate independent. But what is the geometric significance of this property? In the special case where , we get

(A.27) |

So, the invariance of is equivalent to the invariance of the magnitude of vector under transformation.

Let us now investigate the general case. The length squared of in the vector triangle shown in Figure A.6 is

(A.28) |

However, according to the ``cosine rule'' of trigonometry,

(A.29) |

where denotes the length of side . It follows that

(A.30) |

In this case, the invariance of under transformation is equivalent to the invariance of the angle subtended between the two vectors. Note that if then either , , or the vectors and are mutually perpendicular. The angle subtended between two vectors can easily be obtained from the dot product:

(A.31) |

The work performed by a constant force that moves an object through a displacement is the product of the magnitude of times the displacement in the direction of . If the angle subtended between and is then

(A.32) |

The work performed by a non-constant force that moves an object through an infinitesimal displacement in a time interval is . Thus, the rate at which the force does work on the object, which is usually referred to as the power, is , or , where is the object's instantaneous velocity.