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If $\vec{a}'=\hat{i}+\hat{j},\; \vec{b}'=\hat{i}-\hat{j}+2\hat{k}\; and \; \vec{c}'=2\hat{i}+\hat{j}-\hat{k}.$ Then altitude of the parallelepiped formed by the vectors

$\vec{a},\; \vec{b},\; \vec{c}$ having base formed by $\vec{b},\;and\; \vec{c}$ is $\left ( \vec{a},\; \vec{b},\; \vec{c}\; and\; \vec{a}\; ',\vec{b}\: ',\vec{c}\; ',\right)$ are resciprocal systems of vectors

Option: 1
$1$

Option: 2
$\frac{3\sqrt{2}}{2}$

Option: 3
$\frac{1}{\sqrt{6}}$

Option: 4
$\frac{1}{\sqrt{2}}$

Answers (1)

best_answer

Scalar , Dot or Inner Product -

Scalar product of two vector $\vec{A}$ & $\vec{B}$ written as $\vec{A}$ $\cdot$ $\vec{B}$ is a scalar quantity given by the product of magnitude of $\vec{A}$ & $\vec{B}$ and the cosine of smaller angle between them.

$\vec{A}\cdot \vec{B}= A\, B\cdot \cos \Theta$

- wherein

showing representation of scalar products of vectors.

Vector or cross product -

Vector or cross product of two vector $\vec{A}$ & $\vec{B}$ written as $A\times B$ is a single vector whose magnitude is equal to product of magnitude of $\vec{A}$ & $\vec{B}$ and the sine of smaller angle $\Theta$ between them.

$\vec A\times \vec B= A\, B\sin \Theta$

- wherein

Figure 6 shows representation of vector or cross product of vectors.

shows representation of vector or cross product of vectors

Volume of the parallelopiped formed by $\vec{a},'\; \vec{b},'\; \vec{c},'$ is 4

Volume of the parallelopiped formed by $\vec{a},'\; \vec{b},'\; \vec{c},'$ is $\frac{1}{4}$

$\vec{b}\times \vec{c}=\frac{\left ( \vec{c}'\times \vec{a}' \right )\times \vec{c}}{4}=\frac{1}{4}\vec{a}\; '$

$\therefore \; \; \left | \vec{b}\times \vec{c} \right |\frac{\sqrt{2}}{4}=\frac{1}{2\sqrt{2}}$

\ length of altitude = $\frac{1}{4}\times 2\sqrt{2}=\frac{1}{\sqrt{2}}$

Posted by

sudhir.kumar

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