SATHEE: Vectors 2 Question 17

Vectors 2 Question 17

17. If $\vec{b}$ and $\vec{c}$ are any two non-collinear unit vectors and $\vec{a}$ is any vector, then

$(\vec{a} \cdot \vec{b}) \vec{b} + (\vec{a} \cdot \vec{c}) \vec{c} + \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{| \vec{b} \times \vec{c} |^{2}} (\vec{b} \times \vec{c}) = \dots \dots$

(1996, 2M)

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Answer:

Correct Answer: 17. $\sqrt{13}$ sq units 20. True

Solution:

Let $\hat{i}$ be a unit vector in the direction of $\vec{b}, \hat{j}$ in the direction of $\vec{c}$ . Note that $\vec{c} = \hat{j}$

and

$(\vec{b} \times \vec{c}) = | \vec{b} | | \vec{c} | \sin α \hat{k} = \sin α \hat{k}$

where, $\hat{k}$ is a unit vector perpendicular to $\vec{b}$ and $\vec{c}$ .

$\Rightarrow | \vec{b} \times \vec{c} | = \sin α \Rightarrow \hat{k} = \frac{\vec{b} \times \vec{c}}{| \vec{b} \times \vec{c} |}$

Let

$\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$

Now,

$\vec{a} \cdot \vec{b} = \vec{a} \cdot \hat{i} = \hat{i} \cdot (a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}) = a_{1}$

and

$\vec{a} \cdot \vec{c} = \vec{a} \cdot \hat{j} = \hat{j} \cdot (a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}) = a_{2}$

and $\vec{a} \cdot \frac{\vec{b} \times \vec{c}}{| \vec{b} \times \vec{c} |} = \vec{a} \cdot \hat{k} = a_{3}$

$\begin{aligned} ∴ (\vec{a} \cdot \vec{b}) \vec{b} + (\vec{a} \cdot \vec{c}) \vec{c} + \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{| \vec{b} \times \vec{c} |^{2}} (\vec{b} \times \vec{c}) \\ = a_{1} \vec{b} + a_{2} \vec{c} + a_{3} \frac{(\vec{b} \times \vec{c})}{| \vec{b} \times \vec{c} |} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} = \vec{a} \end{aligned}$

Vectors 2 Question 17

17. If $\vec{b}$ and $\vec{c}$ are any two non-collinear unit vectors and $\vec{a}$ is any vector, then

Answer:

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Vectors 2 Question 17

17. If b→ and c→ are any two non-collinear unit vectors and a→ is any vector, then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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17. If $\vec{b}$ and $\vec{c}$ are any two non-collinear unit vectors and $\vec{a}$ is any vector, then