Vectors made visible
Vector projection worksheet
Take vector v = (3, 4) and compare it with the horizontal direction u = (1, 0). The exercise separates the component along that direction from the remaining perpendicular component. All coordinates use the same arbitrary unit. This is a small Euclidean geometry example, not a claim about how a language model represents meaning.
01
Separate the horizontal component
The component of v along u is (3, 0), because u points horizontally and has length one. The remaining vector is (0, 4). Adding those two vectors gives (3, 4), reconstructing the original. The original length is five, from the square root of nine plus sixteen. The component lengths three and four do not add to five; their squares add to twenty-five because the components are perpendicular. Keeping lengths separate from coordinate addition prevents a common misunderstanding in this simple example.
02
Check reconstruction and perpendicularity
Check the dot product of (3, 0) and (0, 4): three times zero plus zero times four equals zero, confirming perpendicularity in this Euclidean setting. Then check reconstruction coordinate by coordinate. These are two different checks: a pair can add to the original without being perpendicular, and a perpendicular pair may not add to the chosen original. Record both results in the worksheet. The exercise is complete when the decomposition satisfies the intended direction, the perpendicular remainder and the reconstruction condition.
03
Change the direction and repeat
For a second case, use the vertical unit direction (0, 1). The projection becomes (0, 4) and the remainder becomes (3, 0), exchanging the earlier roles. A reusable worksheet can hold the original vector, chosen unit direction, projection, remainder, dot-product check and reconstructed vector. If the chosen direction is not a unit vector, the projection calculation needs the direction’s squared length in its denominator. State that condition rather than applying the simplified unit-direction rule to every possible input.