**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

(1)

Observe, we have another vector constructed

(2)

which is orthogonal to . We have a closed form expression for projection, namely

(3)

where is a unit vector. But do we have a closed form expression for ?

**2. Cross-Product.** The cross product of and is

(4)

Note we are using notation from linear algebra writing, recursively,

(5)

where

(6)

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 *

(7)

Another approach would have been to write

(8)

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

(9)

Note this can be negative, and this just tells us information regarding the parallelepiped’s *orientation*.

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.

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