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Algebra of Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors3 Marks
Question
Prove that the given vectors are coplanar:
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $-\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$
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Answer

Given the vectors $\text{P}\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big),\ \text{Q}\big(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}\big)$ and $\text{R}\big(-\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$
We know the three vectors are coplanar if oneof them is expressible as a linear combination of the other two. Let,
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}=\text{x}\big(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}\big)+\text{y}\big(-\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$
$=\hat{\text{i}}(2\text{x}-\text{y})+\hat{\text{j}}(3\text{x}-2\text{y})+\hat{\text{k}}(-\text{x}+2\text{y})$
$\Rightarrow2\text{x}-\text{y}=1,\ 3\text{x}-2\text{y}=1,\ -\text{x}+2\text{y}=1$ [Equating the coefficients of $\hat{\text{i}},\ \hat{\text{j}},\ \hat{\text{k}}$ respectively]
Solving first two of these equation, we get x = 1, y = 1. Clearly these two values satisfy the third equation.
Hence, the given vectors are coplanar.

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