If $\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} a n d \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ are coplanar vectors and $\vec{a} . \vec{c} = 5, \vec{b} ⊥ \vec{c}, t h e n 122 (c_{1} + c_{2} + c_{3})$ is equal to……….

3 Views | Posted 2 weeks ago

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Answered by

Raj Pandey | Contributor-Level 9

2 weeks ago

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

$\begin{array}{l} \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} \\ \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k} \end{array}$

$C o p l n a n a r \Rightarrow | \begin{matrix} 2 & 1 & 3 \\ 3 & 3 & 1 \\ c_{1} & c_{2} & c_{3} \end{matrix} | = 0$

$\begin{array}{l} \Rightarrow - 8 c_{1} + 7 c_{2} + 12 c_{3} = 0 . . . . . . . . (i) \\ \vec{a} . \vec{c} = 5 \Rightarrow 2 c_{1} + c_{2} + 3 c_{3} = 5 . . . . . . . . (i i) \\ \vec{b} . \vec{c} = 0 \Rightarrow 3 c_{1} + 3 c_{2} + c_{3} = 0 . . . . . . . . (i i i) \end{array}$

Solving (i), (ii), (iii)

$C_{1} = \frac{10}{122}, c_{2} = \frac{- 85}{122}, c_{3} = \frac{225}{122}$

$∴ 122 (c_{1} + c_{2} + c_{3}) = 150$

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<mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </mtd> </mtr> </mtable> </math> <math> <mrow> <mi>C</mi> <mi>o</mi> <mi>p</mi> <mi>l</mi> <mi>n</mi> <mi>a</mi> <mi>n</mi> <mi>a</mi> <mi>r</mi> <mo>⇒</mo> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>3</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>3</mn> </mtd> <mtd columnalign="center"> <mn>3</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mtd> <mtd columnalign="center"> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mtd> <mtd columnalign="center"> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math> <math> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mo>⇒</mo> <mo>−</mo> <mn>8</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>7</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mn>2</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>.</mo> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> <mo>=</mo> <mn>5</mn> <mo>⇒</mo> <mn>2</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> <mo>.</mo> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mn>3</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>i</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </math> Solving (i), (ii), (iii) <math> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mn>2</mn> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>8</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>2</mn> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mn>2</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>2</mn> <mn>2</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mo>∴</mo> <mn>1</mn> <mn>2</mn> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <malignmark></malignmark> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mn>5</mn> <mn>0</mn> </mrow> </math>

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If $\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} a n d \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ are coplanar vectors and $\vec{a} . \vec{c} = 5, \vec{b} ⊥ \vec{c}, t h e n 122 (c_{1} + c_{2} + c_{3})$ is equal to……….

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If a → = 2 i ^ + j ^ + 3 k ^ , b → = 3 i ^ + 3 j ^ + k ^ a n d c → = c 1 i ^ + c 2 j ^ + c 3 k ^ are coplanar vectors and a → . c → = 5 , b → ⊥ c → , t h e n 1 2 2 ( c 1 + c 2 + c 3 ) is equal to……….

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If $\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} a n d \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ are coplanar vectors and $\vec{a} . \vec{c} = 5, \vec{b} ⊥ \vec{c}, t h e n 122 (c_{1} + c_{2} + c_{3})$ is equal to……….