Sonoma Academy Multivariable Calculus

Vectors!

What is a vector?

Well, it's magnitude and direction represented by an arrow!

i, j and k are all unit vectors with length 1

i is in the x direction

j is in the y direction

k is in the z direction

Vector Notation:

In 3-Space, vector V starts at the origin and terminates at the point (Vx, Vy, Vz), therefore it has components <Vx, Vy, Vz>. Vx indicated the magnituse of the x-component of the vector, Vy the magnitude of the y-component and Vz the z-component.

<Vx, Vy, Vz> can be translated anywhere along the graph. In other words, two vectors are equal if and only if their corresponding components are equal.

Properties of Vectors:

When adding or subtracting vectors, simply add/subtract the components

Figure 13.2.6: Subtracting vectors is the same as adding the negative vector

Some other properties of Vectors. They essentially follow the same principles of everyday Math!

The resultant vector is always the sum of two vectors!

Here's an example of manipulating vectors

More Properties and Terms:

Finding a vector when given the initial point, P1, and the terminal point, P2:

In both 2 and 3 space, the components of vector P1P2 are found by subtracting P1 from P2

Using Laws of Sines and Cosines to Find the Magnitude and Angle of a Resultant Vector:

Throwback to Trig!! #TBTT :

Red indicates usage of the Law of Cosines. Blue indicates usage of the Law of Sines

Here's an example!!

Normalizing Vectors:

(Watch Out!

Finding the norm and normalizing

are NOT the same!!)

How to find the Norm of a vector (which is the same as the magnitude!)

Normalizing: Creating a unit vector with the same angle as vector V

Heres an example!

We normalized the vector, creating a unit vector with the components 2/3i + 2/3j - 1/3k

Here's Some More Helpful Links:

Graphing Vectors in 3D: http://www.youtube.com/watch?v=51vgIfdBlAk

Properties of Vectors:http://www.youtube.com/watch?v=bGRrZvzNzqY

Properties of vectors (plus a little more we haven't learned yet):http://www.wyzant.com/help/math/calculus/multivariable_vectors/properties_of_vectors

HELLO! Gabe and KT here. Here's a summary of what we learned in Section 13.1 (Monday, August 26th and Tuesday, August 27th).

THE BEGINNING OF 3-SPACE

So now we start dealing with not just (x,y) but (x, y, z).
Note that even when we draw 3-D graphs in 2-D on a paper and it looks like one of the axes are diagonal on the page, it's actually in a right angle in all dimensions.

2-space: quadrants
3-space: OCTANTS
The first octant is where (x, y, z) are all positive.

We have a right hand rule for graphing in 3-space:
Point hand (tips of fingertips) in x-direction
Curl fingers towards y-axis
Thumb subsequently shows z-direction

picture of what our cube looked like on the white board

We did an example where we graphed on our paper a cube of side length 4 with a geometric center at the origin, and were asked to label the vertices. See figure 13.1.3 in the book.

Then we adapted the Pythagorean Theorem into the Distance Formula.

But you can also use the Distance Formula in 3-Space...

And then we took a trip down memory lane to recall Completing the Square.
We were asked to find the center and radius of the circle given by the equation

note: always write equations by order of power, starting with x² and followed by y², etc. This is called standard form

Now we're going to apply Completing the Square to SPHERES.

The parent form of the Sphere equation looks like this:

where x₀, y₀, and z₀ are the point values of the center of the sphere and r is the radius.

Let's do an example:

Find the center and radius of the sphere given by the equation

Then we learned about this word called extrusion.

extrusion: a translation of the 2-D graph in the direction of the "missing variable".

some quick Grapher sketches showing extrusion:

please note that we couldn't figure out how to show the axes labels in Grapher, so you will have to figure out which axis is which on your own! Sorry.