Let alpha, beta, gamma be distinct real numbe | vedclass

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(B) Let $P, Q,$ and $R$ be points with position vectors $\vec{p} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$,$\vec{q} = \beta \hat{i} + \gamma

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Form an equilateral triangle

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(B) Let $P, Q,$ and $R$ be points with position vectors $\vec{p} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$,$\vec{q} = \beta \hat{i} + \gamma \hat{j} + \alpha \hat{k}$,and $\vec{r} = \gamma \hat{i} + \alpha \hat{j} + \beta \hat{k}$.The distance between $P$ and $Q$ is $|\vec{q} - \vec{p}| = \sqrt{(\beta - \alpha)^2 + (\gamma - \beta)^2 + (\alpha - \gamma)^2}$.The distance between $Q$ and $R$ is $|\vec{r} - \vec{q}| = \sqrt{(\gamma - \beta)^2 + (\alpha - \gamma)^2 + (\beta - \alpha)^2}$.The distance between $R$ and $P$ is $|\vec{p} - \vec{r}| = \sqrt{(\alpha - \gamma)^2 + (\beta - \alpha)^2 + (\gamma - \beta)^2}$.Since $|\vec{PQ}| = |\vec{QR}| = |\vec{RP}|$,the points form an equilateral triangle.

Let $\alpha, \beta, \gamma$ be distinct real numbers. The points with position vectors $\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$,$\beta \hat{i} + \gamma \hat{j} + \alpha \hat{k}$,and $\gamma \hat{i} + \alpha \hat{j} + \beta \hat{k}$:

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