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  • Explain Solution R.D.Sharma Class 12 Chapter 23 Scalar or Dot Products Exercise 23.1 Question 43 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 23 Scalar or Dot Products Exercise 23.1 Question 43 Maths Textbook Solution.

Answers (1)

Answer: Proved

Hint: You must know the rules of solving vectors.

Given: If \vec{a} vector \vec{a} is perpendicular to two non-collinear vector \vec{b} and \vec{c} then show that \vec{a} is \perp to every vector in the plane of   \vec{b} and \vec{c} .

Solution: Given that \vec{a} is perpendicular to  \vec{b} and \vec{c}

\vec{a}\cdot \vec{b}=0  and \vec{a}\cdot \vec{c}=0 

Now, let \vec{x}  be any vector in plane of \vec{b} and \vec{c}

Then, \vec{x} is the linear combination of \vec{b} and \vec{c}

\vec{x}=x \vec{b}+y \vec{c} \text { for some } \mathrm{r} \text { and } \mathrm{y}

Now,

\begin{aligned} &\vec{a} \cdot \vec{x}=\vec{a} \cdot(x \vec{b}+y \vec{c}) \\ &\vec{a} \cdot \vec{x}=x(\vec{a} \cdot \vec{b})+y(\vec{a} \cdot \vec{c}) \\ &\vec{a} \cdot \vec{x}=x(0)+y(0) \\ &\vec{a} \cdot \vec{x}=0 \end{aligned}

Thus, \vec{a} \: is \: perpendicular\: to\: \vec{x}

That \: is, \vec{a} \: is \: perpendicular \: to\: every \: vector \: in \: plane \: \vec{b} \: and \: \vec{c}

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