Orthogonality and an angle maximisation

A,B,C have position vectors a,b,c relative to O. AC→=4CB→\overrightarrow{AC}=4\overrightarrow{CB} and ∣a+b∣2=∣a∣2+∣b∣2|\mathbf a+\mathbf b|^2=|\mathbf a|^2+|\mathbf b|^2.

(i) Consider (a+b)⋅(a+b)(\mathbf a+\mathbf b)\cdot(\mathbf a+\mathbf b) to show a and b are perpendicular. marks

(ii) Find the length of projection of c on a in terms of ∣a∣|\mathbf a|. marks

(iii) F is the perpendicular foot from C to OA; f is its position vector. State the geometrical meaning of ∣c×f∣|\mathbf c\times\mathbf f|. mark

(iv) Two points X,Y move along line segments OA,AB respectively such that

OX→=(cos⁡3t)i+(sin⁡3t)j+12k,\overrightarrow{OX}=(\cos3t)\mathbf i+(\sin3t)\mathbf j+\tfrac12\mathbf k, OY→=(sin⁡t)i+(cos⁡t)j−2k,\overrightarrow{OY}=(\sin t)\mathbf i+(\cos t)\mathbf j-2\mathbf k,

where t is real, 0≤t≤2π0\leq t\leq2\pi. Express their scalar product as psin⁡(qt)+rp\sin(qt)+r, with real p,q,r to determine, and find the greatest ∠XOY\angle XOY. marks

[2017/ACJC/Prelim/I/9]

Total marks: 11

Solution

(i)

Perpendicularity

Use (a+b)⋅(a+b)(\mathbf a+\mathbf b)\cdot(\mathbf a+\mathbf b) to show a⊥b\mathbf a\perp\mathbf b.

Explore this idea

Hint

  1. Expand the dot product.

Answer · (i)

a⋅b=0a\cdot b=0.

Explore the idea

A right angle at O

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)

Projection

AC→=4CB→\overrightarrow{AC}=4\overrightarrow{CB}. Find the length of the projection of c\mathbf c on a\mathbf a in terms of ∣a∣|\mathbf a|.

Explore this idea

Hint

  1. Ratio theorem: c=a+4b5\mathbf c=\frac{\mathbf a+4\mathbf b}5.

  2. Project onto a\mathbf a using a⋅b=0\mathbf a\cdot\mathbf b=0.

Answer · (ii)

∣a∣/5|a|/5.

Explore the idea

C and its foot F on OA

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

FF is the foot of the perpendicular from CC to OAOA with position vector f\mathbf f. State the geometrical meaning of ∣c×f∣|\mathbf c\times\mathbf f|.

Hint

  1. ∣c×f∣=OC⋅OFsin⁡∠COF|\mathbf c\times\mathbf f|=OC\cdot OF\sin\angle COF.

Answer · (iii)

Twice the area of triangle OCFOCF.

(iv)

Maximising an angle

OX→=(cos⁡3t,sin⁡3t,12)\overrightarrow{OX}=(\cos3t,\sin3t,\tfrac12) and OY→=(sin⁡t,cos⁡t,−2)\overrightarrow{OY}=(\sin t,\cos t,-2). Express OX→⋅OY→\overrightarrow{OX}\cdot\overrightarrow{OY} as psin⁡(qt)+rp\sin(qt)+r and find the greatest ∠XOY\angle XOY.

Explore this idea

Hint

  1. Use sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B.

  2. Find ∣OX∣|OX| and ∣OY∣|OY|, then cos⁡∠XOY\cos\angle XOY.

Answer · (iv)

p=1,q=4,r=−1p=1,q=4,r=-1; greatest angle cos⁡−1(−4/5)=143.1∘\cos^{-1}(-4/5)=143.1^\circ.

The explicit coordinate paths are circular; the printed fixed-line motion description is inconsistent.

Explore the idea

The angle XOY as t varies

∠XOY = cos⁻¹[(2/5)(sin 4t − 1)] in degrees, for 0 ≤ t ≤ 2π.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Ideas to try
How many times in [0,2π][0,2\pi] does the angle reach its maximum?