Let the vectors
$$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k$$ and $(2 + b)\widehat i + 2b\widehat j + (1 - b)\widehat k$, $a,b,c, \in R$
be co-planar. Then which of the following is true?
Solution
If the vectors are co-planar,<br><br>$$\left| {\matrix{
{a + b + 2} & {a + 2b + c} & { - b - c} \cr
{b + 1} & {2b} & { - b} \cr
{b + 2} & {2b} & {1 - b} \cr
} } \right| = 0$$<br><br>Now, ${R_3} \to {R_3} - {R_2},{R_1} \to {R_1} - {R_2}$<br><br>So, $$\left| {\matrix{
{a + 1} & {a + c} & { - c} \cr
{b + 1} & {2b} & { - b} \cr
1 & 0 & 1 \cr
} } \right| = 0$$<br><br>$= (a + 1)2b - (a + c)(2b + 1) - c( - 2b)$<br><br>$= 2ab + 2b - 2ab - a - 2bc - c + 2bc$<br><br>$= 2b - a - c = 0$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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