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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors4 Marks
Question
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non-zero, non-coplanar vectors, prove that the vector is coplanar:
$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}},\ 7\vec{\text{a}}-8\vec{\text{b}}+9\vec{\text{c}}$ and $3\vec{\text{a}}+20\vec{\text{b}}+5\vec{\text{c}}$
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Answer

We know that,

Three vectors are coplanar if one of the vector can be expressed as the linear combination of other two.

Let,

$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}}\\=\text{x}\big(7\vec{\text{a}}-8\vec{\text{b}}+9\vec{\text{c}}\big)+\text{y}\big(3\vec{\text{a}}+20\vec{\text{b}}+5\vec{\text{c}}\big)$

$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}}\\=7\vec{\text{a}}\text{x}-8\vec{\text{b}}\text{x}+9\vec{\text{c}}\text{x}+3\vec{\text{a}}\text{y}+20\vec{\text{b}}\text{y}+5\vec{\text{c}}\text{y}$

$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}}\\=\big(7\text{x}+3\text{y}\big)\vec{\text{a}}+\big(-8\text{x}+20\text{y}\big)\vec{\text{b}}+\big(9\text{x}+5\text{y}\big)\vec{\text{c}}$

Comparing the LHS and RHS,

7x + 3y = 5 .....(i)

-8x + 20y = 6 .....(ii)

9x + 5y = 7 .....(iii)

For solving (i) and (ii),

Subtract -8 × (i) from 7 × (ii),

$\text{y}=\frac{82}{164}$

$\text{y}=\frac{1}2$

Put $\text{y}=\frac{1}2$ in equation (i),

$7\text{x}+3\text{y}=5$

$7\text{x}+3\Big(\frac{1}2\Big)=5$

$7\text{x}+\frac{3}2=5$

$7\text{x}=\frac{5}1-\frac{3}2$

$7\text{x}=\frac{10-3}2$

$7\text{x}=\frac{7}2$

$\text{x}=\frac{7}{14}$

$\text{x}=\frac{1}{2}$

Now, put $\text{x}=\frac{1}{2}$ and $\text{y}=\frac{1}2$ in equation (iii),

$9\text{x}+5\text{y}=7$

$9\Big(\frac{1}2\Big)+5\Big(\frac{1}2\Big)=7$

$\frac{9}2+\frac{5}2=7$

$\frac{14}2=7$

$7=7$

LHS = RHS

$\therefore$ The value of x, y satisfy equation (iii).

So,

$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}},\ 7\vec{\text{a}}-8\vec{\text{b}}+9\vec{\text{c}},\ 3\vec{\text{a}}+20\vec{\text{b}}+5\vec{\text{c}}$ are coplanar.

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