Vectors in Euclidean Geometry
Given two points A and B...
$\overrightarrow{AB}$ represents the vector 'from A to B'
Suppose $ABCD$ is a parallelogram.
Are $\overline{AB}$ and $\overline{DC}$ the same line segment, or different line segment?
They are different.
A line segment is a geometrical object. In particular a line segment is a set of infinitely many points. This line segment can have properties that are similar to others, but unless it has the exact same infinite many points, they are not equal.
Are $\vec{AB}$ and $\vec{DC}$ the same vector or different vectors?
They are the same.
Vectors can be thought of as properties of line segments.
If two segments have length 6, the same number 6 is used to describe their lengths.
Likewise, the same vector is used to describe the magnitude and direction of two segments with the same magnitude and direction.
- How does this relate to position vectors?
- In this case, the vectors don't have a fixed location, although the line segments being described do.
- Hence, position vectors with the same magnitude and direction are equivalent as position has no effect on the vector
- Vectors are abstract quantities
- Vectors have no location
- Must distinguish between the place where a vector is drawn and its actual properties as a vector.
Addition of vectors in geometry:
$\overrightarrow{PQ}+\overrightarrow{QR} = \overrightarrow{PR}$
When we add arrows with this two letter notation, notice how the Q is the rightmost for the first, and leftmost for the second. They are the same, and in the middle. Hence, you can think of them as cancelling out.
Example:
$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DE}+\overrightarrow{EF}=\overrightarrow{AF}$
- This becomes easier when you think about how the inner letters cancel out.
Abstract Vectors
Vectors can be represented abstractly as mathematical objects which obey certain rules.
They can be notated as lowercase letters in bold
a
Or underline
[insert underlined a]
Note: make sure you don't mistake vectors for variables and vice versa in tests; this can be done by using the correct notation!
Usually, you should write a tilde below.
Regardless of what kind of notation you use, remember each vector carries two pieces of information.
If we wanted to indicate the magnitude of vector a, we denote it via the absolute value signs, i.e. |a|.
e.g. |x| = 6 means the magnitude of vector x is 6 units.
Mathematical Rules for Abstract Vectors
Equality:
- What does it mean for two vectors to be equal, i.e. to be the same?
- "Two vectors are equal if and only if their magnitudes are equal and their directions are equal."
Like:
- Like vectors are either equal or are in the same direction.
- Essentially this means they have to be in the same direction.
Negative:
- If we had a vector a, it follows the vector -a has the same magnitude but opposite(inverse?) direction
- For vector $\overrightarrow{AB}$, then $-\overrightarrow{AB}=\overrightarrow{BA}$.
- i.e. For vector to B from A, the negative of that vector is to A from B.
Scalar Multiplication:
- Given a vector a, what would 2a be?
- For vector a. the vector ka where k is a scalar, is one in the same direction as vector a, but |ka| = k$\times$|a|.
- To multiply by a negative scalar , e.g. think of -2a as 2(-a).
- We know what -a is (see above), and we know how to multiply by a positive scalar, hence combine them together.
- It follows -2a is a vector pointing in the opposite direction of a and has twice the magnitude of a.
Subtracting Vectors:
- Think of a-b as a+(-b).
- From this, it follows one can use vector addition to find the vector of a-b.
- For geometric vectors:
- For $\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{DC}=?$
- We know that $-\overrightarrow{DC}$ is actually $\overrightarrow{CD}$.
- Hence,$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{AD}$.
The Zero Vector:
- The Zero vector is one with magnitude of 0 and an undefined direction.
- Undefined allows for consistency.
- Since its direction is undefined, it's special!!!
- We write as a bold 0, i.e. 0.
- Or you could write it like $\overrightarrow{0}$.
- Does it follow $\overrightarrow{0}=0$?
- Not really. We think of a vector as having two components: magnitude and direction. When we notate a vector, such as $\overrightarrow{0}$. Yet, 0 is simply a magnitude value. Hence, they are different. (Although the direction component is empty, the fact that the vector still has a container is enough to distinguish it from 0.)
