Question

For given vectors \( \vec{a} = -\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} - \hat{j} + \hat{k} \) where \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} = \vec{c} \times \vec{b} \). Then the value of \( (\vec{a}-\vec{b}) \cdot \vec{d} \) is:

• A. \( -35 \)
• B. \( 53 \)
• C. \( -52 \)
• D. \( 25 \)

Hint:

When multiple vector operations are involved, compute step-by-step and keep vectors in component form to avoid sign errors.

Answer 1

The Correct Option is A

To solve this problem, we need to find the value of (→a − →b) · →d, where

→a = −i + j + 2k,  and  →b = 2i − j + k

and →d = →c × →b, with →c = →a × →b.

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Step 1: Calculate →c = →a × →b

→c =

|  i   j   k |
| −1   1   2 |
| 2   −1   1 |

→c = i(1·1 − 2·(−1)) − j((−1)·1 − 2·2) + k((−1)(−1) − 1·2)

→c = 3i + 5j − k

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Step 2: Calculate →d = →c × →b

|  i   j   k |
| 3   5   −1 |
| 2   −1   1 |

→d = i(5·1 − (−1)(−1)) − j(3·1 − (−1)·2) + k(3·(−1) − 5·2)

→d = 4i − 5j − 13k

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Step 3: Calculate (→a − →b) · →d

→a − →b = (−i + j + 2k) − (2i − j + k)

= −3i + 2j + k

(→a − →b) · →d

= (−3)(4) + (2)(−5) + (1)(−13)

= −12 − 10 − 13

= −35

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Final Answer:

(→a − →b) · →d = −35