Let O be the origin. Let OP¯=xi+yj−k , OQ¯=−i+2j+3xk, and OR¯=3i+zj−7k x,y,z∈R,x>0be such that PQ=20, OP¯ is orthogonal to OQ¯, P,Q,R are collinear then the value of x2+y2+z2

OP⊥OQ¯⇒-x+2y-3x=0⇒y=2x…|PQ¯|=20⇒(x+1)2+(2-y)2+(3x+1)2=20⇒14x2-14=0⇒x=±1,x>0 x=1, y=2OP,OQ,OR are collinear xy−1−123x3z−7=0x−14−3xz−y7−9x−1−z−6=0−14−3z−2−2+z+6=0−14−3z+4+z+6=0−2z=4z=−2Therefore, x2+y2+z2=1+4+4=9

Let O be the origin. Let OP¯=xi+yj−k , OQ¯=−i+2j+3xk, and OR¯=3i+zj−7k x,y,z∈R,x>0be such that PQ=20, OP¯ is orthogonal to OQ¯, P,Q,R are collinear then the value of x2+y2+z2

OP⊥OQ¯⇒-x+2y-3x=0⇒y=2x…|PQ¯|=20⇒(x+1)2+(2-y)2+(3x+1)2=20⇒14x2-14=0⇒x=±1,x>0 x=1, y=2OP,OQ,OR are collinear xy−1−123x3z−7=0x−14−3xz−y7−9x−1−z−6=0−14−3z−2−2+z+6=0−14−3z+4+z+6=0−2z=4z=−2Therefore, x2+y2+z2=1+4+4=9

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$\begin{array}{l} Let O be the origin. Let \bar{O P} = x i + y j - k, \bar{O Q} = - i + 2 j + 3 x k, and \bar{O R} = 3 i + z j - 7 k x, y, z \in R, x > 0 \\ be such that |P Q| = \sqrt{20}, \bar{O P} is orthogonal to \bar{O Q}, P, Q, R are collinear then the value of x^{2} + y^{2} + z^{2} \end{array}$

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Detailed Solution

$\begin{array}{l} O P ⊥ \bar{O Q} \Rightarrow - x + 2 y - 3 x = 0 \Rightarrow y = 2 x \dots \\ | \bar{P Q} | = \sqrt{2} 0 \Rightarrow (x + 1)^{2} + (2 - y)^{2} + (3 x + 1)^{2} = 20 \\ \Rightarrow 14 x^{2} - 14 = 0 \Rightarrow x = \pm 1, x > 0 x = 1, y = 2 \\ OP, OQ, OR are collinear \end{array}$

$\begin{matrix} |\begin{matrix} x & y & - 1 \\ - 1 & 2 & 3 x \\ 3 & z & - 7 \end{matrix}| = 0 \\ x (- 14 - 3 x z) - y (7 - 9 x) - 1 (- z - 6) = 0 \\ - 14 - 3 z - 2 (- 2) + z + 6 = 0 \\ - 14 - 3 z + 4 + z + 6 = 0 \\ - 2 z = 4 \\ z = - 2 \end{matrix}$