यदि मूल और बिंदु P (2, 3, 4), Q (1, 2, 3) और R (x, y, z) समतलीय हैं तो:

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  1. x - 2y - z = 0
  2. x + 2y + z = 0
  3. x - 2y + z = 0
  4. 2x - 2y + z = 0

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Option 3 : x - 2y + z = 0
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संकल्पना:

यदि तीन सदिश समतलीय हैं तो उनका अदिश त्रिक गुणनफल शून्य है।

\(\rm \vec a.(\vec b\times \vec c) = 0\)

 

गणना:

दिया है कि मूल और बिंदु P (2, 3, 4), Q (1, 2, 3) और R (x, y, z) समतलीय हैं

\(\rm \vec a = \vec {OR} = {(x, y, z)}\)

\(\rm \vec b = \vec {OP} = {(2, 3, 4)}\)

\(\rm \vec c = \vec {OQ} = {(1, 2, 3)}\)

यहाँ, \(\rm\vec a\) , \(\rm\vec b\) और \(\vec c\) समतलीय हैं

तीन सदिश समतलीय हैं तो यदि उनका अदिश त्रिक गुणनफल शून्य है।

\(\rm \vec a.(\vec b\times \vec c) = 0\)

\(\begin{vmatrix} \rm x & \rm y & \rm z\\ 2&3 &4 \\ 1&2 &3 \end{vmatrix} = 0\)

⇒ x(9 - 8) - y(6 - 4) + z (4 - 3) = 0

⇒ x - 2y + z = 0

इसलिए, यदि मूल और बिंदु P (2, 3, 4), Q (1, 2, 3) और R (x, y, z) समतलीय हैं तो x - 2y + z = 0

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