Vectors - 8.3

Scalar Quantities - quantities characterized by a single real number value called a scalar (area, volume, temperature, and time common examples of scalar quantites)


Directed Line Segment - segment to which a direction has been assigned (referred to as a vector), has both magnitude and direction (not a scalar quantity)

• P is the initial point
• Q is the terminal point
• Arrowhead represents the direction
• The magnitude of vector PQ is the length of the segment PQ and is denoted by ||PQ||

• This vector would jut be called V

Equivalent Vectors - have the same magnitude and direction. The location of the vectors doesn't matter.

• A vector may be traslated from one location to another, provided neither the magnitude nor the direction is changed
• Vectors U and V are equivalent even though they are in different locations

Velocity Vectors can represent many physical concepts

• In this example, vector V (the red vector) is representing the path of a ladybug descending at 100 mph (which determines the magnitude) with a line of flight that makes a 20 degree angle with the horizontal
• V is an example of a velocity vector

Force Vectors - vectors that represent a pull or push of some type

• Vector F represents the upward force 

Displacement - The path of a point as it moves along a segment

• Vector AB represents the path of a point as it moves along the segment AB

Sum of Vectors - any two vectors may be added by placing the initial point of the second vector on the terminal point of the first, then drawing the line segment from the initial point of the first to the terminal point of the second

• This is called the triangle law
• Vector AC is the sum of vectors AB and AC

Resultant force - the single force that produces the same effect as the two combined forces

• Can be found using the parallelogram law
• You form an parallelogram by drawing in imaginary vectors RS and QS
• The force of PS is the resultant force of PR and PQ
• PQ + PR = PS

Scalar Multiple - If m is a scalar and v is a vector, the mv is defined as a vector whose magnitude is |m| times ||v|| and whose direction is either the same as that of v (if m>0) or opposite that of v (if m<0)

• mv is a scalar multiple of v

We generally talk about vectors that have been restricted to an XY plane. Since vectors can be moved around wherever you want as long as the direction and magnitude dont change, we can move the initial point of the vector to (0,0).

Moving the vector here allows the vector to determine an ordered pair

• If our vector has a terminal point A, it can be assigned the coordinates (a₁, a₂) which are the coordinates of terminal point A

There is a one-to-one coorespondance between vectors in an xy-plane and ordered pairs of real numbers

• Allows us to think of vectors as both directed line segments and an ordered pair of real numbers

Instead of using parentheses, vectors expressed by an ordered pair are shown as <a₁, a₂>

• a₁ and a₂ are  called the components of vector <a₁, a₂>
• If A is the point (a₁, a₂), we call vector OA the position vector for <a₁, a₂> or for the point A

Once you understand the basics of vectors, you can learn equations that can be used on vectors when necessary

The magnitude of the vector a = <a₁, a₂>, denoted by ||a||, is given by

The addition of vectors follows the following equation

To find a scalar multiple m of a vector <a₁, a₂>, use 

There is a vector called the zero vector that is denoted as

The negative -a vector of a vector <a₁, a₂> (where m is -1) is

The following is a table that shows all of the properties of vectors. The vectors described are called a, b, and c, along with scalars represented by m and n.

As you can see, the properties of vectors are very similar to those of regular numbers, which makes them easy to remember. 


To subtract vectors, you do the following

There are two special vectors you should remember named i and j

• i = <1,0>
• j = <0,1>

i and j and special because we can use them to denote vectors in yet another way

If we put the above equation on a graph, it will look like this

• Because of its position on the graph, we generally call a1 the horizontal component, and a2 the vertical component. 
• The vector sum a1i + a2j is a linear combination of i and j
• We may regard linear combinations of i and j as algebraic sums

Vector a and angle Θ are defined below (Θ is the entire angle formed by a and the x axis, sorry thats kind of hard to see)

Based on the above diagram, we can confirm the following

Well that's pretty much it for 8.3

I love spending my Wednesday nights writing blogposts about vectors WOOOO

-Corin Murphy