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#specialist#ae

vectors in euclidean geometry §

if you have two points, you can make a vector that starts from point A, B

$\overrightarrow{AB}$ represents the vector ‘from A to B’

Note that $\overrightarrow{AB}$ has vector AB and magnitude $\overline{AB}$ in the case $ABCD$ is a parallelogram

• the lines $\overline{AB}$ and $\overline{DC}$ are distinct and different line segments.
• $\overrightarrow{AB}$ and $\overrightarrow{DC}$ are the same vector.
• think of vectors as properties of directed 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.
  • vectors don’t have a fixed location (but the things they describe might)

addition of vectors in geometry §

addition of vectors in geometry $\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$

furthermore, vectors can be represented abstractly as mathematical objects which obey certain rules.

notations §

vectors can be notated as lowercase letters in bold a

or underlined a

vectors could also have ~ under, above, and an arrow above or under it.

recommended to use ~ underlined, but it’s okay to use another that is conventional.

| v | = magnitude of the vector.

mathematical rules for abstract vectors §

equality §

two vectors are equal if their magnitudes are equal and their directions are equal

• same magnitude, same direction is ‘like’ vectors and equal vectors.

same magnitude, different directions OR same direction, different magnitude and same direction are non equal vectors.

same direction are ‘like’ vectors

negative vectors §

for a vector $a$ the vector $-a$ has the same magnitude but opposite direction.

Hence, if $\overrightarrow{AB}$ is a vector, then $-\overrightarrow{AB}$ is the same vector but the direction is going the other way.

scalar multiplication §

given a vector $a$, what would $2a$ be?

just double the magnitude (only works for positive natural numbers) $2a=a+a$ for a vector $a$ and a positive scalar $k$ (i.e. a positive real number) the vector $ka$ is the vector with the same direction as $a$ and magnitude $k|a|$

to multiply by a negative scalar, e.g. $-2$, think of $-2a$ as $2(-a)$

subtracting vectors §

think of a - b as a + (-b) use addition formula

$\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{DC}=\overrightarrow{AD}$

the zero vector. §

the zero vector 0 has a magnitude of 0 and an undefined direction.

• undefined is consistent with how the rest of maths function.

the zero vector is $\ne$ 0, as a zero only has a magnitude, whilst the zero vector has a magnitude and a direction.