Lecture 8: Review of projections, Introduction to the cross product (Nicholson, Section 4.2)

Alternate Video Access via MyMedia | Video Duration: 49:52 |

Description: Started with a correction to a mistake in previous lecture then did a review of Proj_x (y) and Perp_x (y).

7:00 --- showed that Proj_x (Perp_x (y)) = 0.

25:20 --- started with cross products. The dot product of two vectors works for any two vectors in R^n. The cross product of two vectors only works for vectors in R^3. The cross product of two vectors in R^3 is a vector in R^3.

27:45 --- it’s possible to generalize cross products in some sense --- for example, given 4 vectors in R^5 there’s a way of using them to create a 5^th vector in R^5. This is analogous to a cross product on R^5.

30:00 --- Given a vector equation of a plane, find a scalar equation of the plane. This means that you need to find a normal vector to the plane. This can be done using the cross product of two vectors that are parallel to the plane. (Or it can be done by solving a system of 2 linear equations in 3 unknowns…) Verified that the cross product is orthogonal to the vectors that created it.

42:00 --- presented the formula for how to compute the cross product of two vectors in R^3.

47:00 --- the cross product of a vector with itself is the zero vector. Showed that u x v= - v x u. Proved that u ⋅ (u x v) = 0.