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UPSC NDA Math Model Paper 3

Question : 80 of 120

Marks: +1, -0

Solution:

Given the origin (0, 0, 0) and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planer

\Rightarrow \vec{a} = \overrightarrow{O R} = (x, y, z)

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

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

Here,

\vec{a}, \vec{b}

and

\vec{c}

are co planer

The three vectors are coplanar if their scalar triple product is zero..

\Rightarrow \vec{a} \cdot (\vec{b} \times \vec{c}) = 0

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

\Rightarrow x(9 - 8) - y(6 - 4) + z(4 - 3) = 0

\Rightarrow x - 2y + z = 0

Hence, if the origin and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planar then x - 2y + z = 0

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