Question

The volume of the parallelepiped with its edge by the factors i+j,i+2j,i+j+k is

Answer 1

Step-by-step explanation:

Let the edges of parallelopiped be $\vec{a}=\hat{i}+\hat{j}$, $\vec{b}=\hat{i}+2\hat{j}$ and $\vec{c}=\hat{i}+\hat{j}=\hat{k}$

Then, volume of parallelopiped is given by

$[ \vec{a} \: \: \vec{b} \: \: \vec{c} ]$

$=(\vec{a} \times \vec{b} ). \vec{c}$

$= \{ (\hat{i} \ + \hat{j}) \times ( \hat{i} + 2 \hat{j}) \}. ( \hat{i} + \hat{j} + \hat{k})$

$= \{ \hat{i} \ \times ( \hat{i} + 2 \hat{j}) + \hat{j} \ \times ( \hat{i} + 2 \hat{j}) \}. ( \hat{i} + \hat{j} + \hat{k})$

$= \{ \hat{i} \ \times \hat{i} + 2 \hat{i} \times \hat{j} + \hat{j} \ \times \hat{i} + 2 \hat{j} \times \hat{j} \}. ( \hat{i} + \hat{j} + \hat{k})$

$= \{ 0 + 2 \hat{k} - \hat{k} + 0 \}. ( \hat{i} + \hat{j} + \hat{k})$

$= \hat{k} . ( \hat{i} + \hat{j} + \hat{k})$

$= 1 \: \:$