HSC · Mathematics Extension 2 · Vectors · also VCE Vectors in 3D live from the app

Geometric Proofs Using Vectors

Geometry without the diagrams doing the work: represent points as position vectors and the classical theorems - midpoints, medians, perpendicular diagonals - fall to algebra. Elegant, general and exam-favoured. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

mathematics extension 2 · vectors · geometric proofs using vectorsdrag it · it is yours
the one idea

why this one carries the topic.

A line is one point plus every scalar multiple of a direction; a plane is one point plus every combination of two. That's the whole topic: fix where you start, then say which way you're allowed to move. Every question is just asking where two of these freedoms cross, miss, or coincide.

Represent geometric points as position vectors from the origin, then prove theorems by manipulating these vectors algebraically rather than relying on diagram constructions.

what examiners catch — Students often forget to show that the midpoint formula applies by adding position vectors and dividing by two, or fail to prove collinearity by showing one vector is a scalar multiple of another.
what you leave with

five things, not forty.

what's underneath

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 1 underneath this one, 0 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptgeometric proofs using vectorsRepresent geometric points as position vectors from the origin, then prove theorems by manipulating these vectors algebraically rather than relying on diagram constructions.
sits under itthree-dimensional vector representation and componentsbecause you cannot prove with vectors until points are position vectors
the rest of vectors

3 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

proving inequalitieslive →vector equations of lines and planeslive →scalar product and applications in three dimensionsin the app
this is one of 865

they don't read it. they run it.

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knowherehsc mathematics extension 2geometric proofs using vectors