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➡ Vectors

Vector Definition:: Overall, Vectors let us use NUMBERS to desctibe changes/movements in SPACE, and on the flip side, we can also draw PICTURES of space to better understand our number CALCULATIONS

Vector Definition:: Physics perspective = vectors are arrows, defined by their length and direction, can be translated at will

Vector Definition:: Computer Science perspective = vectors are ordered lists of numbers (vector is prettymuch just a fancy word for list)

Vector Definition:: Mathematician perspective = a vector can be ANYTHING that passes certain criteria (if it adds like normal consistently end-to-end, and scales by scalar multiplication like normal)

Vector Definition:: The examples used here will mostly be in 2D e.g. \(\tbinom{3}{5}\) means 3 along the x-axis, and 5 along the y-axis. But vectors can have many more dimensions (e.g. \(\textstyle \begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}\) for 3D with \(-4\) along the z-axis ), or even use rotations instea of a grid (e.g. see Cylindrical or Spherical bases ).

Vector Definition:: Vectors are represented as combinations of components e.g. \(\tbinom{3}{4}\) is a vector with magnitude 5, that is 3 units long when projected onto the x-axis (vertical shadow) and 4 units long when projected onto the y-axis (horizontal shadow). Projection is simply to measure how much of the length of the whole thing aligns exactly with the subject you’re projecting onto.

Vector Addition:: Each vector represents a certain movement in space. Vectors give instructions for how to travel (what sized steps to take, per axis) from its tail to its tip. So adding vectors means adding each set of steps (per axis) e.g. \(\tbinom{2}{3}\) + \(\tbinom{-4}{1}\) = \(\tbinom{-2}{4}\) . This was 4 individual x-y steps, summarised into 2 resulting x-y steps or 1 resulting vector, called the “resultant vector”, the overall summary of how to get to the end of the addition. So if we add many many vectors, we can summarise the result by just stating the resultant vector (the instructions for how to travel to their end location).

Vector Scaling:: Adding two copies of a vector together v+v=2v gives the same result as just scaling one copy times 2. All its x-axis coordinates get doubled, all its y-axis coordinates … We can also scale vectors down e.g. by multiplying it by ½. Or invert their direction by subtracting rather than adding (same result as multiplying by -1 ). So -½v means pointing v in the opposite direction, and scaling-down how far it goes by ½ in each direction. The quickest way to describe this is “scaling by -½” or “multiplied by the scalar -½”.

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