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Math 126
Lecture 2
Behind us
Last Time
- Intro
- $\mathbf R^3$
- Shapes defined by equations
Quiz Section
- Planes
- Spheres
Questions?
Coming up
vectors, dot products, projections
Read Section 12.2 of the book. We cannot cover everything in lecture or section, but you will need it all for the rest of your lives!
Flaky definition
A vector is an object with two properties: direction and magnitude. For example,
- Displacement from a fixed point
- Velocity
- Acceleration
- Force applied by angry customer
Graphical representation
Draw an arrow:
Graphical representation
Important (yet confusing): two vectors are equivalent if they have the same direction and magnitude. These are all equivalent:
Quick check
Which vectors are equivalent to others in this picture? (Number them 1 through 5 from left to right. For simplicity, they live in a plane.)
Graphical representation
Thus, we will always represent vectors as arrows starting at the origin $(0,0,0)$.
Cheerleading
Does a vector have a position?
I can't hear you!
What does it have?
The magic of vectors
Vectors can be added and scaled.
The magic of vectors
Adding vectors graphically with the triangle rule.
Draw the two vectors to be added using our representation that positions the start at the origin.
The magic of vectors
Adding vectors graphically with the triangle rule.
Translate the second one so that it starts at the end of the first one.
The magic of vectors
Adding vectors graphically with the triangle rule.
Connect the start of the first with the end of the translated second. We end up with purple as red plus blue
The magic of vectors
Check yourself before you wreck yourself
What is the sum of the displacement vector connecting points $p_1$ and $p_2$ and the displacement vector connecting points $p_3$ and $p_4$?
Fun
- Let $A,B,C,D$ be the vertices of a square. Choose a specific example if you want. Compute the sum of vectors $$\vec{AB}+\vec{BC}+\vec{CD}+\vec{DA}.$$
- Let $A_1,A_2,A_3,A_4,A_5,A_6$ be points. Compute the sum $$\vec{A_1A_2}+\vec{A_2A_3}+\vec{A_3A_4}+\vec{A_4A_5}+\vec{A_5A_6}+\vec{A_6A_1}.$$
Numbers breed vectors
We can also describe vectors using numbers.
The vector connecting $A=(a,b,c)$ to $B=(a',b',c')$ is
Note: you must always subtract the coordinates in the same order!
Numbers breed vectors
Any vector is made of three coordinates like that:
$$\mathbf{v}=\langle a,b,c\rangle.$$
Numbers breed vectors
Any vector is made of three coordinates like that:
$$\mathbf v=\langle a,b,c\rangle.$$
This is the same as the vector running from $(0,0,0)$ to the point $(a,b,c)$.
Numbers breed vectors
Any vector is made of three coordinates like that:
$$\mathbf v=\langle a,b,c\rangle.$$
This is the same as the vector running from $(0,0,0)$ to the point $(a,b,c)$.
Its length is
$$||\mathbf v||=\sqrt{a^2+b^2+c^2}$$
Brain massage
- Calculate the length of the vector connecting the point $(0,2,4)$ to the point $(1,-1,1)$.
- Consider the vectors $\langle 1,0,0\rangle$ and $\langle 0,1,0\rangle$. Find $a,b,c$ such that $$\langle 1,0,0\rangle+\langle 0,1,0\rangle=\langle a,b,c\rangle.$$ Try this for another pair of vectors if you finish early. Rinse and repeat.
Numbers breed vectors
Addition and scaling using numbers:
$$\langle a,b,c\rangle +\langle a',b',c'\rangle=\langle a+a', b+b', c+c'\rangle$$
$$\langle 0,3,4\rangle+\langle 1,-1,0\rangle=\langle 1,2,4\rangle$$
$$\gamma\langle a,b,c\rangle=\langle\gamma a,\gamma b,\gamma c\rangle$$
$$3\langle 1,1,2\rangle=\langle 3,3,6\rangle$$
Use it or lose it
- Do the points $$(1,2,3), (2,3,4),(37, 38, 40)$$ lie on a single line in $\mathbf R^3$?
- Find the line containing the largest number of the following points $$(1,0,1), (0,2,0), (1,2,3), (2,2,4), (3,2,5).$$
Next time: dot product!
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