Let <math><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow></math> be two unit vectors such that the angle between them is <math><mrow><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>.</mo></mrow></math> If is the angle between the vectors <math><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover accent="true"><mi>b</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>×</mo><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> then the value of 164cos2 is equal to:

(a^+b^).(a^+2b^+2(a^×b^)),a^.b^=12= 1 + 12+2 + 2 + 0 + 0=3+32|a^+2b^+2(a^×b^)2|=1+4+4(12)+22+0=7+22⇒(2+2)(7+22)cos2θ=9+92+92=27+1822⇒cos2θ=27+1822×118+112=(27+182)(18−112)2(324−242)=486+3242−2972−3962×82164 cos2 (θ)= 90 + 27

Let&nbsp;<math><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow></math>&nbsp;be two unit vectors such that the angle between them is&nbsp;<math><mrow><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>.</mo></mrow></math>&nbsp;If is the angle between the vectors&nbsp;<math><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover accent="true"><mi>b</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mrow><mover accent="true"><mi>a</mi><mo>^</mo></mover><mo>×</mo><mover accent="true"><mi>b</mi><mo>^</mo></mover></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math>&nbsp;then the value of 164cos2 is equal to:

Ask & Answer Question

Maths Class 12th Vector Algebra Maths NCERT Exemplar Solutions Class 12th Chapter Ten

Let $\hat{a} a n d \hat{b}$ be two unit vectors such that the angle between them is $\frac{π}{4} .$ If is the angle between the vectors $(\hat{a} + \hat{b}) a n d (\hat{a} + 2 \hat{b} + 2 (\hat{a} \times \hat{b})),$ then the value of 164cos2 is equal to:

Option 1 -

$90 + 27 \sqrt[]{2}$

Option 2 -

$45 + 18 \sqrt[]{2}$

Option 3 -

$90 + 3 \sqrt[]{2}$

Option 4 -

$54 + 90 \sqrt[]{2}$

5 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$(\hat{a} + \hat{b}) . (\hat{a} + 2 \hat{b} + 2 (\hat{a} \times \hat{b})), \hat{a} . \hat{b} = \frac{1}{\sqrt[]{2}}$

= 1 + $\frac{1}{\sqrt[]{2}} + \sqrt[]{2}$ + 2 + 0 + 0

$= 3 + \frac{3}{\sqrt[]{2}}$

$| \hat{a} + 2 \hat{b} + 2 {(\hat{a} \times \hat{b})}^{2} | = 1 + 4 + 4 (\frac{1}{2}) + 2 \sqrt[]{2} + 0 = 7 + 2 \sqrt[]{2}$

$\Rightarrow (2 + \sqrt[]{2}) (7 + 2 \sqrt[]{2}) c o s^{2} θ = 9 + \frac{9}{2} + 9 \sqrt[]{2} = \frac{27 + 18 \sqrt[]{2}}{2}$

$\Rightarrow c o s^{2} θ = \frac{27 + 18 \sqrt[]{2}}{2} \times \frac{1}{18 + 11 \sqrt[]{2}} = \frac{(27 + 18 \sqrt[]{2}) (18 - 11 \sqrt[]{2})}{2 (324 - 242)}$

$= \frac{486 + 324 \sqrt[]{2} - 297 \sqrt[]{2} - 396}{2 \times 82}$

164 cos² (θ)= 90 + $2$ $7$

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

Let $\hat{a} a n d \hat{b}$ be two unit vectors such that the angle between them is $\frac{π}{4} .$ If is the angle between the vectors $(\hat{a} + \hat{b}) a n d (\hat{a} + 2 \hat{b} + 2 (\hat{a} \times \hat{b})),$ then the value of 164cos2 is equal to:

1 Answer

Learn more about...

Most viewed information

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let a^ and b^ be two unit vectors such that the angle between them is π4. If is the angle between the vectors (a^+b^) and (a^+2b^+2(a^×b^)), then the value of 164cos2 is equal to:

1 Answer

Learn more about...

Most viewed information

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let $\hat{a} a n d \hat{b}$ be two unit vectors such that the angle between them is $\frac{π}{4} .$ If is the angle between the vectors $(\hat{a} + \hat{b}) a n d (\hat{a} + 2 \hat{b} + 2 (\hat{a} \times \hat{b})),$ then the value of 164cos2 is equal to: