The incident ray- reflected ray and the outward drawn normal are denoted by the unit vectors- a- b and c respectively- Then- choose the correct relation for these vectors-

From diagram, a=sinθ î cosθ ĵ b=sinθ î+cosθ ĵ c=ĵ So, a 2(a.c)c =(sinθ î cosθ ĵ) 2[(sinθ î cosθ ĵ).(ĵ)]ĵ=sinθ î cosθ ĵ+2cosθ ĵ=sinθ î+cosθ ĵ=b Hence, opti ...

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Standard XII

Mathematics

Applications of Cross Product

The incident ...

Question

The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors, $\to a, \to b$ and $\to c$ respectively. Then, choose the correct relation for these vectors.

\to b = - \to 2 a + \to c

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\to b = \to a - \to c

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\to b = \to a + 2 \to c

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\to b = \to a - 2 (\to a . \to c) \to c

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Solution

The correct option is D $\to b = \to a - 2 (\to a . \to c) \to c$

From diagram,

$\to a = sin θ^i - cos θ^j$

$\to b = sin θ^i + cos θ^j$

$\to c =^j$

So,
$\to a - 2 (\to a . \to c) \to c$
$= (sin θ^i - cos θ^j) - 2 [(sin θ^i - cos θ^j) . (^j)]^j$ $= sin θ^i - cos θ^j + 2 cos θ^j$ $= sin θ^i + cos θ^j = \to b$

Hence, option $(D)$ is correct.

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Similar questions

{^n}_{1}

is the unit vector along incident ray,

{^n}_{2}

along normal and

{^n}_{3}

is the unit vector along reflected ray, then which of the following must be true?

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\to a =^i -^j

, is incident on a plane mirror having its normal along

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are unit vectors along incident ray in air, refracted ray in a medium, normal respectively and

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Q. Incident ray is along the unit vector

^v

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The normal is along unit vector

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The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors, →a,→b and →c respectively. Then, choose the correct relation for these vectors.

The correct option is D →b=→a−2(→a.→c)→c From diagram, →a=sinθ ^i−cosθ ^j →b=sinθ ^i+cosθ ^j →c=^j So, →a−2(→a.→c)→c =(sinθ ^i−cosθ ^j)−2[(sinθ ^i−cosθ ^j).(^j)]^j=sinθ ^i−cosθ ^j+2cosθ ^j=sinθ ^i+cosθ ^j=→b Hence, option (D) is correct.

The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors, $\to a, \to b$ and $\to c$ respectively. Then, choose the correct relation for these vectors.