Question

The vectors $5a+6b+7c,7a-8b+9c$ and $3a+20b+5c$ are linearly dependent vectors $a,b,c$ being linearly independent vectors.

• A. True
• B. False

Answer 1

The correct option is A True

We know that if these vectors are linearly dependent , then we can express one of them as a linear combination of the other two.

Now let us assume that the given vector are coplanar, then we can write

$5\overrightarrow{a}+6\overrightarrow{b}+7\overrightarrow{c}=l(7\overrightarrow{a}-8\overrightarrow{b}+9\overrightarrow{c})+m(3\overrightarrow{a}+20\overrightarrow{b}+5\overrightarrow{c})$ where $l$ and $m$ are scalars

Comparing the coefficients of $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$

we get

$7l+3m=5$ .......$\left(1\right)$

$-8l+20m=6$ ......$\left(2\right)$

$9l+5m=7$ ......$\left(3\right)$

From equations $\left(1\right)$ and $\left(2\right)$ we get

$20\times \left(1\right)-3\times \left(2\right)$

$140l+60m+24l-60m=100-18$

$\Rightarrow 164l=82$

$\Rightarrow l=\frac{82}{164}=\frac{1}{2}$

Put $l=\frac{1}{2}$ in $\left(1\right)$ we get

$7l+3m=5$

$\Rightarrow 3m=5-7l=5-7\times \frac{1}{2}=\frac{10-7}{2}=\frac{3}{2}$

$\Rightarrow m=\frac{1}{2}$

$\therefore l=m=\frac{1}{2}$which evidently satisfies equation (ii) too.

Hence, The vectors $5a+6b+7c,7a-8b+9c$ and $3a+20b+5c$ are linearly dependent vectors for $a,b,c$ being linearly independent vectors.

Hence the given statement is true.