geometric proofs using vectors — HSC mathematics extension 2, live

Geometric proofs using vectors

Turn every point into a position vector and the classical theorems stop needing a diagram: a midpoint is an average, (b + c)/2, a right angle is a scalar product of zero, and equal sides are equal magnitudes. In a rhombus with side vectors u and v the diagonals are u + v and v − u, so their scalar product is |v|² − |u|² = 0 before anyone measures an angle. The same algebra defines the medians, altitudes, perpendicular bisectors and angle bisectors of a triangle and shows where each triple meets, while the cross product supplies the area |u × v| = |u||v| sin θ. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

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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.

A rhombus built from side vectors u and v, with its diagonals u + v and v − u, their scalar product and the area |u × v|; switch to a triangle to see medians, altitudes, perpendicular bisectors and angle bisectors drawn from their vector definitions, all driven by the angle at A.

what examiners catch — Students often write (b + c)/2 without saying it is the midpoint of BC, or claim two lines are perpendicular because the sketch looks right. What counts in an assessment is the test written in vectors: a scalar product shown to be zero, a scalar multiple shown for collinearity, or λ solved on a line and then interpreted geometrically, such as G dividing the median 2 : 1.

five things, not forty.

A midpoint is an average of position vectors: the midpoint of BC is (b + c)/2, and the median from A is the line a + λ((b + c)/2 − a).

Perpendicular means scalar product zero: in a rhombus AC = u + v and BD = v − u, so AC · BD = |v|² − |u|² = 0 for every angle.

Medians, altitudes, perpendicular bisectors and angle bisectors each have a vector definition, and each triple concurs at G, H, O or I.

The cross product u × v is perpendicular to both u and v, and |u × v| = |u||v| sin θ is the area of the parallelogram they span.

A parallel or collinear claim needs one vector written as a scalar multiple of another; a concurrency claim needs the same point reached from two lines.

Write the proof in vectors rather than coordinates, state each test before using it, and finish with the geometric meaning of the result.

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 5 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 vectorsA rhombus built from side vectors u and v, with its diagonals u + v and v − u, their scalar product and the area |u × v|; switch to a triangle to see medians, altitudes, perpendicular bisectors and angle bisectors drawn from their vector definitions, all driven by the angle at A.

sits under itthree-dimensional vector representation and componentsbecause you cannot prove with vectors until points are position vectors

sits under itplane vectors as arrows and componentsbecause Points as position vectors and arrows between them are the whole language of the proof

sits under itvector addition and scalar multiplesbecause Diagonals as u + v and v − u, and midpoints as averages, are sums and scalar multiples

sits under itscalar product and applications in three dimensionsbecause Perpendicularity is tested with a zero scalar product

sits under itvector product and cross product propertiesbecause The area readout uses |u × v| = |u||v| sin θ

they don't read it. they run it.

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why this one carries the topic.

five things, not forty.

nothing here is a standalone fact.

3 more, same treatment.

they don't read it. they run it.