Question

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b} \cdot \vec{c}=0$ and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$ If $\vec{d}$ be a vector such that $\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}$, then $(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})$ is equal to

• A. $\frac{3}{4}$
• B. $\frac{1}{2}$
• C. $-\frac{1}{4}$
• D. $\frac{1}{4}$

Answer 1

The Correct Option is D

$(\vec{a}\cdot \vec{c})\vec{b}-(\vec{a}\cdot \vec{b})\vec{c}=\frac{\vec{b}-\vec{c}}{2}$
$\vec{a}\cdot \vec{c}=\frac{1}{2},\vec{a}\cdot \vec{b}=\frac{1}{2}$
$\therefore \vec{b}\cdot \vec{d}=\frac{1}{2}$
$(\vec{a}\times \vec{b})\cdot (\vec{c}\times \vec{d})=\vec{a}\cdot (\vec{b}\times (\vec{c}\times \vec{d}))$
$=\vec{a}\cdot ((\vec{b}\cdot \vec{d})\vec{c}-(\vec{b}\cdot \vec{c})\vec{d})$
$=(\vec{a}\cdot \vec{c})(\vec{b}\cdot \vec{d})=\frac{1}{4}$

Answer 2

1. Given \(\vec{a} \times \vec{b} = \frac{\vec{b} - \vec{c}}{2}\): \[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \left(\frac{\vec{b} - \vec{c}}{2}\right) \cdot (\vec{c} \times \vec{d}). \]
2. Using the vector triple product: \[ (\vec{b} - \vec{c}) \cdot (\vec{c} \times \vec{d}) = (\vec{b} \cdot \vec{d})(\vec{c} \cdot \vec{c}) - (\vec{b} \cdot \vec{c})(\vec{c} \cdot \vec{d}). \]
Given \(\vec{b} \cdot \vec{c} = 0\), this simplifies to: \[ (\vec{b} - \vec{c}) \cdot (\vec{c} \times \vec{d}) = (\vec{b} \cdot \vec{d})(\vec{c} \cdot \vec{c}). \]
3. Substitute \(\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b}\) and \(\vec{c} \cdot \vec{c} = |\vec{c}|^2\): \[ (\vec{b} - \vec{c}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{b}) |\vec{c}|^2. \]
4. Substitute back into the original equation: \[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \frac{1}{2} (\vec{a} \cdot \vec{b}) |\vec{c}|^2. \]
From the given conditions, simplify to find: \[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \frac{1}{2}. \]
Thus, the value is \(\frac{1}{2}\). Use vector triple product properties and substitute given conditions step-by-step for precise calculations.