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Coordinate Axes in Three Dimensional Space

The non-param...

Question

The non-parametric vector equation of a plane passing through a point whose position vector is $\to a$ is parallel to $\to u$ and $\to v$ is

[\to r - \to a, \to u, \to v] = 0

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[\to r, \to u, \to v] = 0

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[\to r, \to a, \to u \times \to v]

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[\to a, \to u, \to v] = 0

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Solution

The correct option is D $[\to r - \to a, \to u, \to v] = 0$
Lets recall non-parametric vector equation.
Suppose we want to define a plane that passes through $P_{0}$ and is perpendicular to n.
NOTE: There will be only one such plane because fixing a direction fixes the orientation of our plane.

Lets define r as set of all points which lie on the plane.
Now, observe that $r - r_{0}$ lies on the plane itself because both
$P, P_{0}$ lie on the plane.
Thus, equation of plane becomes
$(r - r_{0}) . n = 0$ (because $c o s 90 = 0$ )

In the given case , $n = u \times v$ and $r_{0} = a$
Replacing in our equation we get, $(r - a) . (u \times v) = 0$
$[r - a, u, v] = 0$

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