SATHEE: Vectors 1 Question 24

Vectors 1 Question 24

25. Find 3-dimensional vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}$ satisfying

$\begin{aligned} \vec{v_{1}} \cdot {\vec{v}}_{1} = 4, \vec{v_{1}} \cdot {\vec{v}}_{2} = - 2, \vec{v_{1}} \cdot {\vec{v}}_{3} = 6 \\ {\vec{v}}_{2} \cdot {\vec{v}}_{2} = 2, {\vec{v}}_{2} \cdot {\vec{v}}_{3} = - 5, {\vec{v}}_{3} \cdot {\vec{v}}_{3} = 29 \end{aligned}$

(2001, 5M)

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Answer:

Correct Answer: 25. (b)

Solution:

We have, $| {\vec{v}}_{1} | = 2, | {\vec{v}}_{2} | = \sqrt{2}$ and $| {\vec{v}}_{3} | = \sqrt{29}$

If $θ$ is the angle between ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ , then

$\begin{aligned} 2 \sqrt{2} \cos θ & = - 2 \\ \Rightarrow \cos θ & = - \frac{1}{\sqrt{2}} \\ \Rightarrow & θ & = 135^{\circ} \end{aligned}$

Since, any two vectors are always coplanar and data is not sufficient, so we can assume ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ in $x - y$ plane.

$\begin{aligned} {\vec{v}}_{1} = 2 \hat{i} \\ {\vec{v}}_{2} = - \hat{i} + \hat{j} \end{aligned}$

and

$\vec{v_{3}} = α \hat{i} + β \hat{j} + γ \hat{k}$

Since, ${\vec{v}}_{3} \cdot {\vec{v}}_{1} = 6 = 2 α \Rightarrow α = 3$

Also, ${\vec{v}}_{3} \cdot {\vec{v}}_{2} = - 5 = - α \pm β \Rightarrow β = \pm 2$

and ${\vec{v}}_{3} \cdot {\vec{v}}_{3} = 29 = α^{2} + β^{2} + γ^{2} \Rightarrow γ = \pm 4$

Hence, ${\vec{v}}_{3} = 3 \hat{i} \pm 2 \hat{j} \pm 4 \hat{k}$

पिछला

अगला

Vectors 1 Question 24

25. Find 3-dimensional vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}$ satisfying

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Vectors 1 Question 24

25. Find 3-dimensional vectors v→1,v→2,v→3 satisfying

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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25. Find 3-dimensional vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}$ satisfying