NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

let $a = 2 \hat{i} + 2 \hat{j} + \hat{k}$ and $b$ be another vector such that $a \cdot b = 14$ and $a \times b = 3 \hat{i} + \hat{j} - 8 \hat{k}$ . Find the vector $b$ .

Choose the correct answer:

Explanation

To find vector $b$ , we use the property of the vector triple product: $a \times (a \times b) = (a \cdot b) a - (a \cdot a) b$ .

1. Given values:

$a = 2 \hat{i} + 2 \hat{j} + \hat{k}$
$a \cdot b = 14$
$a \times b = 3 \hat{i} + \hat{j} - 8 \hat{k}$

2. Calculate magnitude terms:

$a \cdot a = ∣ a ∣^{2} = 2^{2} + 2^{2} + 1^{2} = 4 + 4 + 1 = 9$

3. Calculate $a \times (a \times b)$ :

Using the cross product formula for $a \times (3 \hat{i} + \hat{j} - 8 \hat{k})$ :

$a \times (a \times b) = \hat{i} 23 am p; \hat{j} am p; 2 am p; 1 am p; \hat{k} am p; 1 am p; - 8$

$= \hat{i} (- 16 - 1) - \hat{j} (- 16 - 3) + \hat{k} (2 - 6)$

$= - 17 \hat{i} + 19 \hat{j} - 4 \hat{k}$

4. Solve for $b$ using the identity:

$- 17 \hat{i} + 19 \hat{j} - 4 \hat{k} = (14) (2 \hat{i} + 2 \hat{j} + \hat{k}) - 9 b$

$- 17 \hat{i} + 19 \hat{j} - 4 \hat{k} = 28 \hat{i} + 28 \hat{j} + 14 \hat{k} - 9 b$

$9 b = (28 \hat{i} + 28 \hat{j} + 14 \hat{k}) - (- 17 \hat{i} + 19 \hat{j} - 4 \hat{k})$

$9 b = 45 \hat{i} + 9 \hat{j} - 18 \hat{k}$

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

Conclusion:

The vector $b$ is $5 \hat{i} + \hat{j} - 2 \hat{k}$ . The correct option is (C).

Choose the correct answer:

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