Equal radii make a midpoint perpendicular

A and B have position vectors a,b\mathbf a,\mathbf b relative to O, with ∣a∣=∣b∣|\mathbf a|=|\mathbf b|. P with position vector p\mathbf p lies on AB such that b⋅p=a⋅p\mathbf b\cdot\mathbf p=\mathbf a\cdot\mathbf p.

(i) Show AB is perpendicular to OP. marks

(ii) Determine the position vector of D in terms of a,b, where D is O reflected about AB. marks

(iii) Give the geometrical meaning of ∣a×b∣|\mathbf a\times\mathbf b|. mark

[2010/NJC/Prelim/I/3]

Total marks: 5

Solution

(i)

Perpendicularity

∣a∣=∣b∣|\mathbf a|=|\mathbf b| and b⋅p=a⋅p\mathbf b\cdot\mathbf p=\mathbf a\cdot\mathbf p with PP on ABAB. Show AB⊥OPAB\perp OP.

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Hint

  1. Rearrange to (b−a)⋅p=0(\mathbf b-\mathbf a)\cdot\mathbf p=0.

Answer · (i)

(b−a)⋅p=0(b-a)\cdot p=0, so AB⊥OPAB\perp OP.

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An isosceles triangle OAB

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(ii)

Reflection

Find OD→\overrightarrow{OD} where DD is the reflection of OO in ABAB.

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Hint

  1. With ∣a∣=∣b∣|\mathbf a|=|\mathbf b|, PP is the midpoint of ABAB.

  2. DD satisfies OD→=2OP→\overrightarrow{OD}=2\overrightarrow{OP}.

Answer · (ii)

d=a+bd=a+b.

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The rhombus OADB

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(iii)

Geometrical meaning

Give the geometrical meaning of ∣a×b∣|\mathbf a\times\mathbf b|.

Hint

  1. ∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta.

Answer · (iii)

Area of the rhombus OADBOADB.

A≠BA\ne B; for a genuine rhombus the sides must also be nonparallel.