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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors5 Marks
Question
Show that the four points A, B, C and D with the position vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ and $\vec{\text{d}}$ respectively are coplanar if and only if $3\vec{\text{a}}-2\vec{\text{b}}+\vec{\text{c}}-2\vec{\text{d}}=\vec0$.
✓

Answer

Necessary Condition: Firstly, let $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are coplanar vectors. Then, one of them is expressible as a linear combination of the other two.Let $\vec{\text{c}}=\text{x}\vec{\text{a}}+\text{y}\vec{\text{b}}$ for some scalars x, y. Then,
$\Rightarrow\ \text{l}\vec{\text{a}}+\text{m}\vec{\text{b}}+\text{n}\vec{\text{c}}=\vec0$, where l = x, m = y, n = -1.
Thus, if $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are coplanar vectors, then there exists a scalars l, m, n not all zero simultaneously satisfying $\text{l}\vec{\text{a}}+\text{m}\vec{\text{b}}+\text{n}\vec{\text{c}}=\vec0$ where l, m, n are not all zero simultaneously.
Sufficient Condition: Let $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are three scalars such that there exists scalars l, m, n not all zero simultaneously satisfying $\text{l}\vec{\text{a}}+\text{m}\vec{\text{b}}+\text{n}\vec{\text{c}}=\vec0$. We have to prove that $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are coplanar vectors.
Now,
$\text{l}\vec{\text{a}}+\text{m}\vec{\text{b}}+\text{n}\vec{\text{c}}=\vec0$
$\Rightarrow\text{n}\vec{\text{c}}=-\text{l}\vec{\text{a}}-\text{m}\vec{\text{b}}$
$\Rightarrow\ \vec{\text{c}}=\Big(\frac{-\text{l}}{\text{n}}\Big)\vec{\text{a}}+\Big(\frac{-\text{m}}{\text{n}}\Big)\vec{\text{b}}$
$\vec{\text{c}}$ is a linear combination of $\vec{\text{a}}\text{ and }\vec{\text{b}}$.
$\vec{\text{c}}$ lies in a plane $\vec{\text{a}}\text{ and }\vec{\text{b}}$.
Hence, $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are coplanar vectors.

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