SATHEE: Vectors 2 Question 10

Vectors 2 Question 10

10. If the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ from the sides $B C, C A$ and $A B$ respectively of a $△ A B C$ , then

(2000, 2M)

(a) $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0$

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

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

(d) $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \vec{0}$

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Answer:

Correct Answer: 10. (b)

Solution:

By triangle law, $\vec{a} + \vec{b} + \vec{c} = \vec{0}$

Taking cross product by $\vec{a}, \vec{b}, \vec{c}$ respectively,

$\vec{a} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{a} \times \vec{0} = \vec{0}$

$\Rightarrow \vec{a} \times \vec{a} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = \vec{a}$

$\Rightarrow \vec{a} \times \vec{b} = \vec{c} \times \vec{a} [∵ \vec{a} \times \vec{a} = \vec{0}]$

$Similarly, \vec{a} \times \vec{b} = \vec{b} \times \vec{a}$

$∴ \vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}$

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