We continued from last lecture, by looking at the geometrical interpretation of the dot-product. We showed that the dot product can be interpreted as the length of one vector times the signed projection of the other vector onto the line defined by the first vector. [Transparancy 1]
Next, we demonstrated the usefulness of vectors in solving geometry problems, using a couple of examples from the textbook, section 1.7.
The first example [Transparancies 1 and 2] looked at the object formed by joining the successive mid-points of any quadrilateral. We proved that this object will always be a parallelogram. The proof, using vectors, is actually quite general, because it applies even when the 4 vertices of the quadrilateral are not confined to the same plane in space!
The second example [Transparancies 3 and 4] looked at the two line segments from a vertex of a parallelogram to the midpoints of the opposite sides. It was proven that these two line segments trisect a diagonal of the parallelogram. This example was a bit more complicated than the first one, and involved introducing a couple of parameters, "s" and "t".
We next looked at the equation of a line in space [Transparancy 5]. This can be written as a simple vector equation, using a parameter "t". One can also find a non-parametric form for the equations of a line in space, simply by eliminating "t". (For more details, see section 1.8 of the textbook.)
Finally, we looked at the equation of a plane [Transparancy 6]. It is easy to write in terms of the dot-product. We will talk more about this in Monday's lecture. You could read section 1.10 to prepare.
Here are the scanned-in transparancies, in full-colour JPEG format:
