SATHEE: Vectors 3 Question 27

Vectors 3 Question 27

27. If $\vec{u}, \vec{v}, \vec{w}$ are three non-coplanar unit vectors and $α, β, γ$ are the angles between $\vec{u}$ and $\vec{v}, \vec{v}$ and $\vec{w}, \vec{w}$ and $\vec{u}$ respectively and $\vec{x}, \vec{y}, \vec{z}$ are unit vectors along the bisectors of the angles $α, β, γ$ respectively. Prove that

$[\vec{x} \times \vec{y} \vec{y} \times \vec{z} \vec{z} \times \vec{x}] = \frac{1}{16} [\vec{u} \vec{v} \vec{w}]^{2} \sec^{2} \frac{α}{2} \sec^{2} \frac{β}{2} \sec^{2} \frac{γ}{2}$

(2003, 4M)

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Answer:

Correct Answer: 27. $\frac{146}{17}$

Solution:

$\vec{x} = \frac{\vec{u} + \vec{v}}{| \vec{u} + \vec{v} |} = \frac{1}{2} \sec \frac{α}{2} (\vec{u} + \vec{v})$

$\begin{array}{r} \vec{y} = \frac{\vec{w} + \vec{v}}{| \vec{v} + \vec{w} |} = \frac{1}{2} \sec \frac{β}{2} (\vec{v} + \vec{w}) \\ \vec{z} = \frac{\vec{w} + \vec{u}}{| \vec{w} + \vec{u} |} = \frac{1}{2} \sec \frac{γ}{2} (\vec{w} + \vec{u}) \end{array}$

Since, $[\vec{x} \times \vec{y} \vec{y} \times \vec{z} \vec{z} \times \vec{x}] = [\vec{x} \vec{y} \vec{z}]^{2}$

[from Eq. (i)]

$= \frac{1}{64} \sec^{2} \frac{α}{2} \cdot \sec^{2} \frac{β}{2} \cdot \sec^{2} \frac{γ}{2} [\vec{u} + \vec{v} \vec{v} + \vec{w} \vec{w} + \vec{u}]^{2}$

and $[\vec{u} + \vec{v} \vec{v} + \vec{w} \vec{w} + \vec{u}] = 2 [\vec{u} \vec{v} \vec{w}]$

$∴ [\vec{x} \times \vec{y} \vec{y} \times \vec{z} \vec{z} \times \vec{x}]$

$\begin{aligned} = \frac{1}{64} \sec^{2} \frac{α}{2} \cdot \sec^{2} \frac{β}{2} \cdot \sec^{2} \frac{γ}{2} \cdot 4 [\vec{u} \vec{v} \vec{w}]^{2} \\ = \frac{1}{16} \cdot [\vec{u} \vec{v} \vec{w}]^{2} \sec^{2} \frac{α}{2} \cdot \sec^{2} \frac{β}{2} \cdot \sec^{2} \frac{γ}{2} \end{aligned}$

Vectors 3 Question 27

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Vectors 3 Question 27

27. If u→,v→,w→ are three non-coplanar unit vectors and α,β,γ are the angles between u→ and v→,v→ and w→,w→ and u→ respectively and x→,y→,z→ are unit vectors along the bisectors of the angles α,β,γ respectively. Prove that

Answer:

Engineering Preparation

Medical Preparation

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Important Resources

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Syllabus

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