Vectors

#Calc-2#Math#Vectors Scalars: (numbers) 5, -13, $\pi$ etc

• have magnitude only
• no sense of directions Compare to vectors, for example, I am traveling at 65 mph north
• There’s a sense of magnitude - 65 mph
• and a sense of direction - north

Mathematically, vectors are represented by arrows

written as vector $\overrightarrow{OP}$ , put the point you start at, then the point you end at, and put an arrow on top vector $\overrightarrow{OP}=<1,2>=1\vec{i}+2\vec{j}$ ^ this is how far to move right, the other number is how far to move up i indicates a vector in the x direction, j a vector in the y direction, and k (not here yet) in the z direction

in general, <destination x - starting x, destination y - starting y>

What about vector $\overrightarrow{PO}$

$\overrightarrow{PO}=<-1,-2>$

Note: $\overrightarrow{PO}=-1\ast \overrightarrow{OP}$ In general, multiplying a vector by -1 reverses its direction while keeping the magnitude the same

Multiplying by scalars §

very simple to do

$2\overrightarrow{OP}=<2\ast 1,2\ast 2>=<2,4>$

This changes the magnitude (length of the vector) by a factor equal to the scalar

$-3\overrightarrow{OP}$ reverses OP’s direction and lengthens it by a factor of 3

Finding Magnitude §

To find the magnitude (length) of vector OP, written as $|\overrightarrow{OP}|$

There’s a right angle triangle in there! We know the base and the height, so the magnitude is

$|\overrightarrow{OP}|=\sqrt{(1{)}^2+(2{)}^2}=\sqrt{5}$

Generalized, if $\vec{u}=<a,b>$, then the magnitude is $|\vec{u}|=\sqrt{a^2+b^2}$

Moving Vectors §

Copies of the same vector at different locations are equal Each of these $\vec{v}$ is called a position vector (gives the relative difference between start and end)

Even if a vector is moved, they are the same as long as they have the same magnitude and direction

Two Vectors are equal if they have the same magnitude and the same direction

Vector Operations §

Adding and Subtracting vectors §

Algebraically, very easy

$\vec{u}=<1,2>\vec{v}=<0,3>$ $\vec{u}+\vec{v}=<1+0,2+3>=<1,5>$ $\vec{u}-\vec{v}=<1-0,2-3>=<1,-1>$

$2\vec{u}-3\vec{v}=<2,4>-<0,9>=<2,-5>$

Graphically, things get more interesting

When adding, keep first vector the same and move the second vectors tail to the head of the first. Create a new vector from the tail of the first vector to the head of the second to get the final vector result.

This is called the Triangle Rule

Another way called the Parallelogram Rule It basically does the same thing, just looks a bit different.

Subtraction graphically looks like doing $\vec{u}+(-\vec{v})$

Multiplying, just look at scalars above its that §

Dot Product and Cross Product §

Unit Vectors §

These are vectors with a magnitude of 1

The general unit vectors are (the parentheses are only for 3D vectors, as is the $\vec{k}$)

• $\vec{i}=<1,0,(0)>$
• $\vec{j}=<0,1,(0)>$
• $\vec{k}=<0,0,1>$

A unit vector parallel to $|\overrightarrow{PQ}|$ is $\frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}=<\frac{3}{\sqrt{50}},\frac{4}{\sqrt{50}},\frac{5}{\sqrt{50}}>$ Another one is $-\frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}$