Mathematics

Problem

Let

\mathbf{a}

and

\mathbf{b}

be two non-zero vectors. a. (i) State the greatest and least possible values of

\,\mathbf{a}\cdot\mathbf{b}\,

in terms of

|\mathbf{a}|

and

|\mathbf{b}|

. b. (ii) Prove that the sign of

\,\mathbf{a}\cdot\mathbf{b}\,

determines whether the angle between

\mathbf{a}

and

\mathbf{b}

is acute, right, or obtuse. c. (iii) Give one numerical example of each of the three cases in part (ii) by choosing suitable 3-dimensional vectors, and verify your conclusion in each case.

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