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# Prove that a necessary and sufficient condition for three vectors to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that l → a + m → b + n → c = → 0 . - CBSE (Science) Class 12 - Mathematics

ConceptScalar Triple Product of Vectors

#### Question

Prove that a necessary and sufficient condition for three vectors $\vec{a}$, $\vec{b}$, $\vec{c}$  to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that $l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .$

#### Solution

$\text{ Necessary Condition: First let }\vec{a} , \vec{b} , \vec{c} \text{ be three coplanar vectors . Then one of them is expressable as a linear combination of the other two . }$
$\text{Let }\vec{c} = x \vec{a} + y \vec{b}\text{ for some scalars x, y . }$
$\text{Then, }\vec{c} = x \vec{a} + y \vec{b}\text{ for some scalars x, y }$
$\Rightarrow l \vec{a} + m \vec{b} + n \vec{c} = 0,\text{ where }l = x, m = y, n = - 1$
$\text{ Thus, if }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors, then there exists scalars l, m, n such that l }\vec{a} + m \vec{b} + n \vec{c} = 0\text{ where l, m, n are all non zero simultaneously . }$

$\text{ Sufficient Condition: Let }\vec{a} , \vec{b} , \vec{c}\text{ be three vectors such that there exists scalars l, m, n not all zero simulataneously }$

$\text{ satisfying l }\vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\text{ We have tp prove that }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors . }$

$\text{Now, l }\vec{a} + m \vec{b} + n \vec{c} = \vec{0}$

$\Rightarrow n \vec{c} = - l \vec{a} - m \vec{b}$

$\Rightarrow \vec{c} = \left( - \frac{1}{n} \right) \vec{a} + \left( - \frac{m}{n} \right) \vec{b}$

$\Rightarrow \vec{c}\text{ is a linear combination of }\vec{a} \text{ and }\vec{b} .$

$\text{ Hence }\vec{a} , \vec{b} , \vec{c}\text{ are coplanar vectors }.$

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Solution Prove that a necessary and sufficient condition for three vectors to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that l → a + m → b + n → c = → 0 . Concept: Scalar Triple Product of Vectors.
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