First slide

Introduction to 3-D Geometry

Question

$Three distinct points P (3 u^{2}, 2 u^{3}), Q (3 v^{2}, 2 v^{3}) and$ $R (3 w^{2}, 2 w^{3}) are collinear then -$

Moderate

A

$u v + v w + w u = 0$
B

$u v + v w + w u = 3$
C

$u v + v w + w u = 2$
D

$u v + w w + w u = 1$

Solution

$|\begin{matrix} 3 u^{2} & 2 u^{3} & 1 \\ 3 v^{2} & 2 v^{3} & 1 \\ 3 w^{2} & 2 w^{3} & 1 \end{matrix}| = 0$
$R_{1} \to R_{1} - R_{2} and R_{2} \to R_{2} - R_{3}$
$\Rightarrow |\begin{matrix} u + v & u^{2} + v^{2} + vu & 0 \\ v + w & v^{2} + w^{2} + vw & 0 \\ w^{2} & w^{3} & 1 \end{matrix}| = 0$
$E x a p a n d t h e d e t e r m i n a n t$
$(u + v) (v^{2} + w^{2} + v w) - (v + w) (u^{2} + v^{2} + u v) = 0 \Rightarrow u v^{2} + u w^{2} + u v w + v^{3} + v w^{2} + v^{2} w - v u^{2} - v^{3} - u v^{2} - u^{2} w - v^{2} w - u v w = 0 \Rightarrow u w (w - u) + v w (w - v) = 0$
$\Rightarrow uv + vw + wu = 0$

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