A vector \( \vec{d} \) is equally inclined to three vectors \( \vec...

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A vector \( \vec{d} \) is equally inclined to three vectors \( \vec{a}=\hat{i}-\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+\hat{j} \) and \( \vec{c}=3 \hat{j}-2 \hat{k} \). Let
\( \mathrm{P} \) \( \vec{x}, \vec{y} \) and \( \vec{z} \) be three vectors in the plane of \( \vec{a}, \vec{b} ; \vec{b}, \vec{c} ; \vec{c}, \vec{a} \), respectively. Then

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(1) \( \vec{x} \cdot \vec{d}=-1 \)
(2) \( \vec{y} \cdot \vec{d}=1 \)
(3) \( \vec{z} \cdot \vec{d}=0 \)
(4) \( \vec{r} \cdot \vec{d}=0, \quad \) where \( \vec{r}=\lambda \vec{x}+\mu \vec{y}+\delta \vec{z} \)
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