lesson
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Every time a video game character dashes across your screen, the physics engine translates that motion using an arrow in space called a vector.
To let a computer calculate motion, rotation, or scaling, we write that geometric arrow as a single, organized list of numbers: a column matrix.
๐Interactive/animated comparison of a 2D arrow on a grid and its column matrix representation. On the left, a 2D Cartesian plane showing a vector arrow starting at origin (0,0) and pointing to (3, 4) with labeled horizontal step +3 (in blue) and vertical step +4 (in green). On the right, the column matrix [3; 4] with 3 highlighted in blue and labeled 'x-component (horizontal)' and 4 in green labeled 'y-component (vertical)'. Clear, clean card layout with soft borders.
How do we bridge the visual world of arrows with the algebraic world of matrices?
From Arrows to Matrices
A matrix is a rectangular grid of numbers organized into rows and columns, described by its dimensions rowsรcolumns.
A 2D geometric vector represents a shift of x units horizontally and y units vertically, written as a 2ร1 column matrix:
v=[xyโ]
The top row always tracks the horizontal change along the x-axis, while the bottom row tracks the vertical change along the y-axis.
What happens when a vector starts somewhere other than the origin?
Vectors Between Two Points
To find the vector AB from initial point A(x1โ,y1โ) to terminal point B(x2โ,y2โ), subtract the starting coordinates from the ending coordinates:
AB=[x2โโx1โy2โโy1โโ]