Affine collinearity and a triangle area

Relative to O, A,B,D have position vectors a,b,d. OACB is a parallelogram and ∠OAC=2π/3\angle OAC=2\pi/3 radians.

(i) Given a is a unit vector and ∣b∣=4|\mathbf b|=4, find the length of projection of OC onto OA. marks

(ii) Given λa+μb+d=0\lambda\mathbf a+\mu\mathbf b+\mathbf d=\mathbf0 and λ+μ+1=0\lambda+\mu+1=0, show A,B,D are collinear. marks If μ=4\mu=4, find area OBD as k∣a×b∣k|\mathbf a\times\mathbf b|, determining k. marks

[2017/VJC/Prelim/II/2]

Total marks: 9

Solution

(i)

Projection in a parallelogram

OACBOACB is a parallelogram, ∣a∣=1|\mathbf a|=1, ∣b∣=4|\mathbf b|=4, ∠OAC=2π3\angle OAC=\frac{2\pi}3. Find the length of the projection of OC→\overrightarrow{OC} onto OA→\overrightarrow{OA}.

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Hint

  1. ∠AOB=π−2π3=π3\angle AOB=\pi-\frac{2\pi}3=\frac\pi3.

  2. c=a+b\mathbf c=\mathbf a+\mathbf b.

Answer · (i)

33.

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The parallelogram OACB

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Collinearity

Affine combinations

Given λa+μb+d=0\lambda\mathbf a+\mu\mathbf b+\mathbf d=\mathbf 0 with λ+μ+1=0\lambda+\mu+1=0, show A,B,DA,B,D are collinear.

Hint

  1. Substitute λ=−1−μ\lambda=-1-\mu.

  2. Show DA→\overrightarrow{DA} is a multiple of AB→\overrightarrow{AB}.

Answer · Collinearity

d=a−μ(b−a)d=a-\mu(b-a), so DD lies on line ABAB.

Area of triangle OBD

Cross-product area

If μ=4\mu=4, find the area of triangle OBDOBD as k∣a×b∣k|\mathbf a\times\mathbf b|.

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Hint

  1. λ=−5\lambda=-5, so d=5a−4b\mathbf d=5\mathbf a-4\mathbf b.

  2. Area =12∣b×d∣=\frac12|\mathbf b\times\mathbf d|.

Answer · Area of triangle OBD

k=5/2k=5/2.

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D on line AB (μ = 4)

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