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Solve the following.
If the angle between a⃗ = -3î + xĵ + k̂ and b⃗ = xî + 2xĵ + k̂ is acute and the angle between…
If the angle between $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ is acute and the angle between $\vec{b}$ and the x-axis lies between $\frac{\pi}{2}$ and $\pi$. Find the range of values for $x$.
Answer
Answer
Official answer key$\vec{a}\cdot\vec{b} > 0$ as the angle is acute.
$\Rightarrow -3x + 2x^2 + 1 > 0$
$\Rightarrow (x-1)(2x-1) > 0$
$\Rightarrow x < \frac{1}{2}$ or $x > 1$
$\vec{b}\cdot\hat{i} < 0$ as the angle lies between $\frac{\pi}{2}$ and $\pi$
$\Rightarrow x < 0$
Final answer: $x < 0$