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Intercept Made by Circle on Axes

a,b,c are thr...

Question

$a, b, c$ are three non-coplanar vectors. Then the equation $\to r = (1 - p - q) \to a + p \to b + q \to c$ represents :

A

A straight line

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B

A plane

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C

A plane passing through the origin

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D

A sphere

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Solution

The correct option is C A plane
Let $\to r$ be also represented in the form of three non-coplanar vectors $\to a, \to b, \to c$ .
Let $\to r = x \to a + y \to b + z \to c$
Hence, $x \to a + y \to b + z \to c = (1 - p - q) \to a + p \to b + q \to c$
Comparing RHS to LHS gives us
$x = 1 - p - q$
$y = p$
$z = q$
Then $x = 1 - y - z$
$x + y + z = 1$
This represents an equation of a plane where the co-ordinate axes are $\to a$ , $\to b$ and $\to c$ , and also the plane does not pass through origin.

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