Vectors 5 Question 9

9. Incident ray is along the unit vector v^ and the reflected ray is along the unit vector w^. The normal is along unit vector a^ outwards. Express w^ in terms of a^ and v^.

(2005,4 M)

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Answer:

Correct Answer: 9. w^=v^2(a^.v^)a^

Solution:

  1. Since, v^ is unit vector along the incident ray and w^ is the unit vector along the reflected ray.

Hence, a^ is a unit vector along the external bisector of v^ and w^.

w^v^=λa^

On squaring both sides, we get

1+1w^v^=λ222cos2θ=λ2λ=2sinθ

where, 2θ is the angle between v^ and w^.

Hence, w^v^=2sinθa^=2cos(90θ)a^=(2a^v^)a^

w^=v^2(a^v^)a^



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