Question

Area of parallelogram formed by adjacent sides as the vectors A=3i + 2j and B=2j - 4k is

Answer 1

Explanation:

it is always perpendicular to its plane. so, area of parallelogram formed by two adjacent sides a and b = cross product of a and b. hence area of parallelogram is 27 sq unit

Answer 2

Given:

Adjacent sides of parallelogram :-

$\bf{\vec{A}=4\hat{i}+2\hat{j}}$

$\bf{\vec{B}=2\hat{j}-4\hat{k}}$

To Find:

Area of given parallelogram

Solution:

We know that,

• For a parallelogram having adjacent sides $\vec{a}$ and $\vec{b}$

$\bf\purple{Area\:of\:Parallelogram=\mid \vec{a}\times\vec{b}\mid}$

Now,

For given parallelogram

$\rm{Area=\mid \vec{A}\times\vec{B}\mid}$

So,

$\rm{\vec{A}\times\vec{B}=\left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\3&2&0\\0&2&-4\end{array}\right|}$

$\rm{\vec{A}\times\vec{B}=(-8-0)\hat{i}-(-12-0)\hat{j}+(6-0)\hat{k}}$

$\rm{\vec{A}\times\vec{B}=-8\hat{i}+12\hat{j}+6\hat{k}}$

Now,

$\rm{\mid\vec{A}\times\vec{B}\mid=\sqrt{(-8)^{2}+12^{2}+6^{2}}}$

$\rm{\mid\vec{A}\times\vec{B}\mid=\sqrt{64+144+36}}$

$\rm{\mid\vec{A}\times\vec{B}\mid=\sqrt{244}}$

$\rm{\mid\vec{A}\times\vec{B}\mid=\sqrt{4\times61}}$

$\rm{\mid\vec{A}\times\vec{B}\mid=2\sqrt{61}}$

Therefore,

$\longrightarrow\rm\pink{Area\:of\:Parallelogram=2\sqrt{61}}$

Hence, the area of given parallelogram is 2√61