SATHEE: Vectors 4 Question 14

Vectors 4 Question 14

14. If the vectors $\vec{b}, \vec{c}, \vec{d}$ are not coplanar, then prove that the vector $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) + (\vec{a} \times \vec{c}) \times (\vec{d} \times \vec{b})$

$+ (\vec{a} \times \vec{d}) \times (\vec{b} \times \vec{c}) is parallel to \vec{a}$

$(1994, 4 M)$

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Answer:

Correct Answer: 14. 5

Solution:

Considering first part $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})$

Let

$\vec{c} \times \vec{d} = \vec{e}$

$(\vec{a} \times \vec{b}) \times \vec{e} = (\vec{a} \cdot \vec{e}) \vec{b} - (\vec{b} \cdot \vec{e}) \vec{a}$ $[∵ (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}]$

$= \vec{a} \cdot (\vec{c} \times \vec{d}) \vec{b} - \vec{b} \cdot (\vec{c} \times \vec{d}) \vec{d}$

$= [\vec{a} \vec{c} \vec{d}] \vec{b} - [\vec{b} \vec{c} \vec{d}] \vec{a}$

Similarly,

$\begin{aligned} (\vec{a} \times \vec{c}) \times (\vec{d} \times \vec{b}) & = [\vec{a} \vec{d} \vec{b}] \vec{c} - [\vec{c} \vec{d} \vec{b}] \vec{a} \\ = [\vec{a} \vec{d} \vec{b}] \vec{c} - [\vec{b} \vec{c} \vec{d}] \vec{a} \end{aligned}$

Also, $(\vec{a} \times \vec{d}) \times (\vec{b} \times \vec{c}) = - (\vec{b} \times \vec{c}) \times (\vec{a} \times \vec{d})$

$\begin{aligned} = (\vec{b} \times \vec{c}) \times (\vec{d} \times \vec{a}) & = [\vec{b} \vec{d} \vec{a}] \vec{c} - [\vec{c} \vec{d} \vec{a}] \vec{b} \\ = [\vec{a} \vec{d} \vec{b}] \vec{c} - [\vec{a} \vec{c} \vec{d}] \vec{b} \end{aligned}$

From Eqs. (i), (ii) and (iii),

$(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) + (\vec{a} \times \vec{c}) \times (\vec{d} \times \vec{b}) + (\vec{a} \times \vec{d}) \times (\vec{b} \times \vec{c})$

$= [\vec{a} c \vec{d}] \vec{b} - [\vec{b} \vec{c} \vec{d}] \vec{a} + [\vec{a} \vec{d} \vec{b}] \vec{c} - [\vec{b} \vec{c} \vec{d}] \vec{a} - [\vec{a} \vec{d} \vec{b}] \vec{c}$

$- [\vec{a} \vec{c} \vec{d}] \vec{b} = - 2 [\vec{b} \vec{c} \vec{d}] \vec{a}$

$∴$ Parallel to $\vec{a}$ .

Vectors 4 Question 14

14. If the vectors $\vec{b}, \vec{c}, \vec{d}$ are not coplanar, then prove that the vector $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) + (\vec{a} \times \vec{c}) \times (\vec{d} \times \vec{b})$

Answer:

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Vectors 4 Question 14

14. If the vectors b→,c→,d→ are not coplanar, then prove that the vector (a→×b→)×(c→×d→)+(a→×c→)×(d→×b→)

Answer:

Engineering Preparation

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Important Resources

Other Resources

Syllabus

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Mentorship

NCERT Exercise Video Solution

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14. If the vectors $\vec{b}, \vec{c}, \vec{d}$ are not coplanar, then prove that the vector $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) + (\vec{a} \times \vec{c}) \times (\vec{d} \times \vec{b})$