Find \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , when \[\vec{a} = 2 \hat{i} - 3 \hat{j} , \vec{b} = \hat{i} + \hat{j} - \hat{k} \text{ and } \vec{c} = 3 \hat{i} - \hat{k}\]

Given : \[ \vec{a} = 2 \hat{ i } - 3 \hat{j} \] \[ \vec{b} =\hat{ i} + \hat{j} - \hat{k} \] \[ \vec{c} = 3 \hat{i}- \hat{k} \] \[ \therefore \vec{a} \times \vec{b} = \left( 2\hat{ i} - 3 \hat{j} \right) \times \left( \hat{i} + \hat{j} - \hat{k} \right)\] \[ = 2 \hat{k} + 2 \hat{j} + 3 \hat{k} + 3 \hat{i} \] \[ = 3 \hat{i} + 2 \hat{j} + 5 \hat{k} \] \[\left( \vec{a} \times \vec{b} \right) . \vec{c} = \left( 3\hat{ i} + 2 \hat{j} + 5\hat{ k} \right) . \left( 3 \hat{i} - \hat{k} \right)\] \[ = 9 - 5 = 4 . . . \left( 1 \right)\] Now, \[\left[ \vec{a} \vec{b} \vec{c} \right] = \left( \vec{a} \times \vec{b} \right) . \vec{c} \] \[ = 4 \left[ \text{Using} \left( 1 \right) \right]\]

Find [ → a → B → C ] , When → a = 2 ^ I − 3 ^ J , → B = ^ I + ^ J − ^ K and → C = 3 ^ I − ^ K - Mathematics

प्रश्न

Find $[\vec{a} \vec{b} \vec{c}]$ , when $\vec{a} = 2 \hat{i} - 3 \hat{j}, \vec{b} = \hat{i} + \hat{j} - \hat{k} and \vec{c} = 3 \hat{i} - \hat{k}$

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उत्तर

Given :

$\vec{a} = 2 \hat{i} - 3 \hat{j}$

$\vec{b} = \hat{i} + \hat{j} - \hat{k}$

$\vec{c} = 3 \hat{i} - \hat{k}$

$∴ \vec{a} \times \vec{b} = (2 \hat{i} - 3 \hat{j}) \times (\hat{i} + \hat{j} - \hat{k})$

$= 2 \hat{k} + 2 \hat{j} + 3 \hat{k} + 3 \hat{i}$

$= 3 \hat{i} + 2 \hat{j} + 5 \hat{k}$

$(\vec{a} \times \vec{b}) . \vec{c} = (3 \hat{i} + 2 \hat{j} + 5 \hat{k}) . (3 \hat{i} - \hat{k})$

$= 9 - 5 = 4 . . . (1)$

Now,

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

$= 4 [Using (1)]$

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Scalar Triple Product of Vectors

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Q 2.1 Q 1.2 Q 2.2

अध्याय 26: Scalar Triple Product - Exercise 26.1 [पृष्ठ १६]

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अध्याय 26 Scalar Triple Product
Exercise 26.1 | Q 2.1 | पृष्ठ १६

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Scalar Triple Product of Vectors

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Find [ → a → B → C ] , When → a = 2 ^ I − 3 ^ J , → B = ^ I + ^ J − ^ K and → C = 3 ^ I − ^ K - Mathematics

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