If <math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>6</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>3</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mo>,</mo> </mrow> </math> <math> <mrow> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mn>3</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>2</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> and <math> <mrow> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> are linearly dependent vectors and <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mo stretchy="true">→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mrow> <mn>6</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math> then sum of all possible value of (a + b) is <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>β</mi> <mtext> </mtext> <mtext> </mtext> <mo>∈</mo> <mtext> </mtext> <mtext> </mtext> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>

[ a → b → c → ] = 0 | 1 6 3 3 2 1 α + 1 β − 1 1 | = 0 8 β − 2 4 = 0 ⇒ β = 3 | c → | = 6 ( α + 1 ) 2 + ( β − 1 ) 2 + 1 = 6 ⇒ ( α + 1 ) 2 = 1 a = 1 = 1, 1 ⇒ α = − 2 , 0 + = 1, 3

Ask & Answer Question

Maths Class 12th Vector Algebra

If $\vec{a} = \hat{i} + 6 \hat{j} + 3 \hat{k},$ $\vec{b} = 3 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{c} = (α + 1) \hat{i} + (β - 1) \hat{j} + \hat{k}$ are linearly dependent vectors and $| \vec{c} | = \sqrt[]{6}$ then sum of all possible value of (a + b) is $(α, β \in R)$

Option 1 -

Option 2 -

Option 3 -

Option 4 -

2 Views | Posted 3 weeks ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

3 weeks ago

Correct Option - 4

Detailed Solution:

$[\vec{a} \vec{b} \vec{c}] = 0$

$| \begin{matrix} 1 & 6 & 3 \\ 3 & 2 & 1 \\ α + 1 & β - 1 & 1 \end{matrix} | = 0$

$8 β - 24 = 0 \Rightarrow β = 3$

$| \vec{c} | = \sqrt[]{6}$

${(α + 1)}^{2} + {(β - 1)}^{2} + 1 = 6 \Rightarrow {(α + 1)}^{2} = 1$

a = 1 = 1, 1 $\Rightarrow α = - 2, 0$

+ = 1, 3

<math> <mrow> <mrow> <mo> [</mo> <mrow> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math> <math> <mrow> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>6</mn> </mtd> <mtd columnalign="center"> <mn>3</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>3</mn> </mtd> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mtd> <mtd columnalign="center"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math> <math> <mrow> <mn>8</mn> <mi>β</mi> <mo>−</mo> <mn>2</mn> <mn>4</mn> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mi>β</mi> <mo>=</mo> <mn>3</mn> </mrow> </math> <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mrow> <mn>6</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math> <math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>6</mn> <mo>⇒</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math> a = 1 = 1, 1 <math> <mrow> <mo>⇒</mo> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>, </mo> <mn>0</mn> </mrow> </math> <math><mrow></mrow></math>+ = 1, 3

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

If $\vec{a} = \hat{i} + 6 \hat{j} + 3 \hat{k},$ $\vec{b} = 3 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{c} = (α + 1) \hat{i} + (β - 1) \hat{j} + \hat{k}$ are linearly dependent vectors and $| \vec{c} | = \sqrt[]{6}$ then sum of all possible value of (a + b) is $(α, β \in R)$

1 Answer

Learn more about...

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

If a → = i ^ + 6 j ^ + 3 k ^ , b → = 3 i ^ + 2 j ^ + k ^ and c → = ( α + 1 ) i ^ + ( β − 1 ) j ^ + k ^ are linearly dependent vectors and | c → | = 6 then sum of all possible value of (a + b) is ( α , β ∈ R )

1 Answer

Learn more about...

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

If $\vec{a} = \hat{i} + 6 \hat{j} + 3 \hat{k},$ $\vec{b} = 3 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{c} = (α + 1) \hat{i} + (β - 1) \hat{j} + \hat{k}$ are linearly dependent vectors and $| \vec{c} | = \sqrt[]{6}$ then sum of all possible value of (a + b) is $(α, β \in R)$