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  • Explain solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 9 maths textbook solution

Explain solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 9 maths textbook solution

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Answer: Vectors are coplanar

Hint: Prove and express that vectors can be coplanar if we can express them as the linear combination.

\RightarrowGiven, three vectors \overrightarrow{a},\overrightarrow{b} & \overrightarrow{c} to be proved as coplanar

Now “Necessary condition”

To be coplanar vectors must be expressed as linear combination of other two.

If \overrightarrow{a},\overrightarrow{b} & \overrightarrow{c}  has to be coplanar.

There exists,\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}            .....(1)

Where x and y be some scalars.

Hence, l\overrightarrow{c}+m\overrightarrow{b}+n\overrightarrow{c}=0       ....(2)

Comparing (1) and (2)

We can say, l=x

m = y

c = -1

For l\overrightarrow{c}+m\overrightarrow{b}+n\overrightarrow{c}=0 to be coplanar they must satisfy this.Wherel,m and n are non-zero simultaneously.

“Sufficient Condition”

If we suppose that\overrightarrow{a},\overrightarrow{b} & \overrightarrow{c}  be three vectors satisfying l\overrightarrow{c}+m\overrightarrow{b}+n\overrightarrow{c}=0

Where l,m and n not all zero simultaneously as scalars.

So from, l\overrightarrow{c}+m\overrightarrow{b}+n\overrightarrow{c}=0

\begin{aligned} &n \vec{c}=-l \vec{a}-m \vec{b} \\ &\vec{c}=\left(\frac{-l}{n}\right) \vec{a}+\left(\frac{-m}{n}\right) \vec{b} \end{aligned}

Thus, \overrightarrow{c} can be written as linear combination of \overrightarrow{a} & \overrightarrow{b}where \left ( \frac{-l}{n} \right )  and \left ( \frac{-m}{n} \right ) be some scalars.

Hence, \overrightarrow{a},\overrightarrow{b} & \overrightarrow{c}  are coplanar vectors.

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