1. Last time we ended with discussing how to project a vector onto another. So if we consider projecting onto , we can write this as
Observe, we have another vector constructed
which is orthogonal to . We have a closed form expression for projection, namely
where is a unit vector. But do we have a closed form expression for ?
2. Cross-Product. The cross product of and is
Note we are using notation from linear algebra writing, recursively,
Remark 1. Observe this implies , , and .
Remark 2. The cross-product takes two vectors, and produces a third vector. It does not produce a scalar (a number, unlike the dot product).
Pop quiz: let and be vectors. Is ?
Example 1. Consider and . What is ?
Solution: we find
Another approach would have been to write
and used the cross-product’s anticommutativity to do the calculations.
3. Parallelogram Area. Consider three distinct points , , and . We can construct a parallelogram, as in the following diagram:
We see that , then the area of the parallelogram is .
4. Parallelepiped Volume. If we work in 3-space, and we have a six-sided region whose sides are each parallelograms, we call this region a parallelepiped. Observe that we only need 3 vectors to specify the vertices: , , and . Then we consider , , , and for the remaining vertices. What is the volume of this region?
Lets draw a diagram:
Lets first consider the face described by and . We see the parallelepiped may be considered as a “stack” of such faces, whose height is given by the third vector . Then we see the area of the face is shaded in the diagram, and algebraically it’s given by , and this produces a vector whose magnitude is the area of the face. When we “dot” this with , it’s intuitively taking the product of the “area of a face” () and the “height of the parallelepiped” () producing the volume
Note this can be negative, and this just tells us information regarding the parallelepiped’s orientation.