Discussion
A vector \(\vec{A} = \vec{i} + x\vec{j} + 3\vec{k}\) is rotated through an angle and is also doubled in magnitude resulting in \(\vec{B} = 4\vec{i} + (4x - 2)\vec{j} + 2\vec{k}\). An acceptable value of \(x\) is
Answer: B
An acceptable value is x = 2.
Rotation does not change length, so doubling gives \(|\vec{B}| = 2|\vec{A}|\), that is \(|\vec{B}|^{2} = 4|\vec{A}|^{2}\).
\[ \begin{aligned} 16 + (4x - 2)^{2} + 4 &= 4(1 + x^{2} + 9) \\ 16x^{2} - 16x + 24 &= 4x^{2} + 40 \\ 3x^{2} - 4x - 4 &= 0 \end{aligned} \]
\[ \begin{aligned} (x - 2)(3x + 2) &= 0 \\ x &= 2 \text{ or } -\tfrac{2}{3} \end{aligned} \]
Check x = 2: \(|\vec{A}|^{2} = 14\) and \(|\vec{B}|^{2} = 16 + 36 + 4 = 56 = 4 \times 14\). The values 1, 3 and 4/3 do not satisfy the equation.
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