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NSWMaths Extension 1Quick questions

Vectors (ME-V1)

Quick questions on Geometric proofs with vectors: parallel, perpendicular, midpoint and ratio properties

15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is position vectors and triangle setup?
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Choose an origin OO. Each point PP has a position vector p=OP\mathbf{p} = \mathbf{OP}.
What is parallelism?
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u\mathbf{u} and v\mathbf{v} are parallel iff v=λu\mathbf{v} = \lambda \mathbf{u} for some scalar λ\lambda. If λ>0\lambda > 0, same direction; if λ<0\lambda < 0, opposite direction. Equality of vectors (u=v\mathbf{u} = \mathbf{v}) is a stronger statement (parallel and equal length).
What is perpendicularity?
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u\mathbf{u} and v\mathbf{v} are perpendicular iff u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0.
What is midpoint?
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The midpoint of ABAB has position vector
What is section formula?
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The point dividing ABAB internally in the ratio m:nm : n (so APPB=mn\frac{AP}{PB} = \frac{m}{n}) has position vector
What is parallelogram criterion?
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ABCDABCD is a parallelogram iff AB=DC\mathbf{AB} = \mathbf{DC} (opposite sides equal and parallel). Equivalently, the diagonals bisect each other: midpoint of ACAC equals midpoint of BDBD.
What is rectangle criterion?
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A parallelogram is a rectangle iff one of its angles is right, i.e., AB⋅AD=0\mathbf{AB} \cdot \mathbf{AD} = 0.
What is standard tactic?
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To prove a geometric statement:
What is diagonals of a parallelogram bisect each other?
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ABCDABCD is a parallelogram, so AB=DC\mathbf{AB} = \mathbf{DC}, that is b−a=c−d\mathbf{b} - \mathbf{a} = \mathbf{c} - \mathbf{d}, equivalently a+c=b+d\mathbf{a} + \mathbf{c} = \mathbf{b} + \mathbf{d}.
What is cosine rule from vectors?
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∣a−b∣2=(a−b)⋅(a−b)=∣a∣2−2a⋅b+∣b∣2|\mathbf{a} - \mathbf{b}|^2 = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = |\mathbf{a}|^2 - 2 \mathbf{a} \cdot \mathbf{b} + |\mathbf{b}|^2.
What is section in given ratio?
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PP divides ABAB in the ratio 1:31 : 3. Find PP in terms of a\mathbf{a} and b\mathbf{b}.
What is perpendicular diagonals imply a rhombus?
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Show that if a parallelogram has perpendicular diagonals, it is a rhombus.
What is equating direction with vector?
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Two vectors are parallel if they are scalar multiples; they are equal only if they are the same direction AND the same magnitude.
What is confusing midpoint with section?
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Midpoint is the special case of the section formula with ratio 1:11 : 1.
What is skipping the dot-product test?
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Perpendicularity is tested algebraically with u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0, not by "looking at the diagram". :::

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