Mathematics

Mathematics

Problem

Let O be the origin. Let

\overline{OP} = x\hat{i} + y\hat{j} - \hat{k}

and

\overline{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}

,

x, y \in \mathbb{R}

,

x > 0

, be such that

|\overline{PQ}| = \sqrt{20}

and the vector

\overline{OP}

is perpendicular to

\overline{OQ}

. If

\overline{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}

,

z \in \mathbb{R}

, is coplanar with

\overline{OP}

and

\overline{OQ}

, then the value of

x^2 + y^2 + z^2

is equal to (1) 7 (2) 9 (3) 2 (4) 1

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