(D) The magnitude of a vector remains invariant under rotation. Therefore, $|\vec{v}|^2 = |\vec{v}'|^2$.$|\vec{v}|^2 = 3^2 + 4^2 + (-5)^2 = 9 + 16 + 2

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(D) The magnitude of a vector remains invariant under rotation. Therefore, $|\vec{v}|^2 = |\vec{v}'|^2$.$|\vec{v}|^2 = 3^2 + 4^2 + (-5)^2 = 9 + 16 + 25 = 50$.$|\vec{v}'|^2 = (a + 1)^2 + (-3)^2 + 5^2 = (a + 1)^2 + 9 + 25 = (a + 1)^2 + 34$.Equating the two: $(a + 1)^2 + 34 = 50$.$(a + 1)^2 = 16$.$a + 1 = \pm 4$.Case $1$: $a + 1 = 4 \implies a = 3$.Case $2$: $a + 1 = -4 \implies a = -5$.Thus, the possible values of $a$ are $3$ or $-5$.

Class 12 Mathematics Vector Algebra Basic , Modulus and Algebra of vectors

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If a vector $\vec{v} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated about the origin, its magnitude remains constant. If the new vector is $\vec{v}' = (a + 1)\hat{i} - 3\hat{j} + 5\hat{k}$, find the possible values of $a$.

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A
$5 \text{ or } 3$
B
$5 \text{ or } -3$
C
$4 \text{ or } -2$
D
$-5 \text{ or } 3$

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If a vector $\vec{v} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated about the origin, its magnitude remains constant. If the new vector is $\vec{v}' = (a + 1)\hat{i} - 3\hat{j} + 5\hat{k}$, find the possible values of $a$.

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