If {a} = 2{i} - 3{j} + {k} , {b} = {i} - {j} + 2{k} and {c} = 2{i} + {j} + {k} are three vectors, then ≤ft| ({a} × {b}) × {c} | =

Correct Answer is : 1139∣a⃗×(b⃗×c⃗)∣{ 11/39 } ≤ft| {a} × ({b} × {c}) |3911a×(b×c)

{ {39} }{ {11} } ≤ft| {a} × ({b} × {c}) |

{ 11/39 } ≤ft| {a} × ({b} × {c}) |

{11} ≤ft| {a} × ({b} × {c}) |

Correct Answer is : 1139∣a⃗×(b⃗×c⃗)∣{ 11/39 } ≤ft| {a} × ({b} × {c}) |3911a×(b×c)

Test Index

TS EAMCET 04 May 2019 Shift 2 Solved Paper

Section: Mathematics

Question : 34 of 160

Marks: +1, -0

\vec{a}

2\hat{i} - 3\hat{j} + \hat{k}

\vec{b}

\hat{i} - \hat{j} + 2\hat{k}

\vec{c}

2\hat{i} + \hat{j} + \hat{k}

\left| (\vec{a} \times \vec{b}) \times \vec{c} \right|

$\left| \vec{a} \times (\vec{b} \times \vec{c}) \right|$
$\frac{ \sqrt{39} }{ \sqrt{11} } \left| \vec{a} \times (\vec{b} \times \vec{c}) \right|$
$\sqrt{ \frac{11}{39} } \left| \vec{a} \times (\vec{b} \times \vec{c}) \right|$
$\sqrt{11} \left| \vec{a} \times (\vec{b} \times \vec{c}) \right|$

Solution:

Consider the vectors,

\overline{a}=2\overline{i}-3\overline{j}+\overline{k}

\overline{b}=\overline{i}-\overline{j}+2\overline{k}

And

\overline{c}=2\overline{i}+\overline{j}+\overline{k}

\left| (\overline{a} \times \overline{b}) \times \overline{c} \right| \;\; = \left| (\overline{c} \cdot \overline{a}) \overline{b} - (\overline{c} \cdot \overline{b}) \overline{a} \right|

\;\; = \left| (2\overline{i} - 2\overline{j} + 4\overline{k}) - (\overline{6}\overline{i} - 9\overline{j} + 3\overline{k}) \right|

\;\; = \left| -4\overline{i} + 7\overline{j} + \overline{k} \right|

\;\; = \sqrt{66}

And,

\left| \overline{a} \times (\overline{b} \times \overline{c}) \right| = \left| (\overline{a} \cdot \overline{c}) \overline{b} - (\overline{a} \cdot \overline{b}) \overline{c} \right|

= \left| (2\overline{i} - 2\overline{j} + 4\overline{k}) - (14\overline{i} + 7\overline{j} + 7\overline{k}) \right|

= \left| -12\overline{i} - 9\overline{j} - 3\overline{k} \right|

= \sqrt{234}

Therefore,

\; \frac{ \left| (\overline{a} \times \overline{b}) \times \overline{c} \right| }{ \left| \overline{a} \times (\overline{b} \times \overline{c}) \right| } = \sqrt{ \; \frac{66}{234} }

\left| (\overline{a} \times \overline{b}) \times \overline{c} \right| = \sqrt{ \; \frac{11}{39} } \left| \overline{a} \times (\overline{b} \times \overline{c}) \right|

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