Projection followed by line reflection

Relative to origin O, A and B have position vectors a=3i+j+3k\mathbf a=3\mathbf i+\mathbf j+3\mathbf k and b=5i−4j+3k\mathbf b=5\mathbf i-4\mathbf j+3\mathbf k.

(i) Find the length of the projection of OA on OB. marks

(ii) Hence or otherwise find the position vector of C on OB such that AC is perpendicular to OB. marks

(iii) Find a vector equation of the reflection of line AB in line AC. marks

[2010/ACJC/Prelim/I/9]

Total marks: 7

Solution

(i)

Length of a projection

Find the length of the projection of OA→\overrightarrow{OA} on OB→\overrightarrow{OB}.

Hint

  1. Use ∣a⋅b∣∣b∣\frac{|\mathbf a\cdot\mathbf b|}{|\mathbf b|}.

Answer · (i)

222\sqrt2.

(ii)

Perpendicular foot

Find CC on OBOB with AC⊥OBAC\perp OB.

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Hint

  1. CC is the foot of the perpendicular from AA to OBOB: OC→=25OB→\overrightarrow{OC}=\frac25\overrightarrow{OB}.

Answer · (ii)

C=(2,−8/5,6/5)C=(2,-8/5,6/5).

Explore the idea

Slide C along OB

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)

Reflecting a line

Find a vector equation of the reflection of line ABAB in line ACAC.

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Hint

  1. CC is the midpoint of BB and its image B′B'.

  2. The reflected line is AB′AB'.

Answer · (iii)

r=(3,1,3)+t(20,1,18)r=(3,1,3)+t(20,1,18), t∈Rt\in\mathbb R.

Explore the idea

Move B and watch its reflection in AC

Drag a coordinate slice or adjust x, y and z. The foot and reflected segment update together; rotate the 3D scene to inspect them.

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