Question

Find the area of the parallelogram whose adjacent sides are determined by the vectors

$\vec{a} = \hat{i} - \hat{j} + 3\hat{k}\\ \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$

Answer 1

We know that if Vectors $\vec{p}$ and $\vec{q}$ are the Adjacent sides of a Parallelogram, then Area of the Parallelogram is given by : $|\:\vec{p}\:\times\vec{q}\:|$

Given that : a = i - j + 3k and b = 2i - 7j + k are the Adjacent sides of the Parallelogram ⇒ Area of the Parallelogram will be : I a × b I

First let us find a × b :

$a\:\times\:b = \left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\1&-1&3\\2&-7&1\end{array}\right|$

a × b = i(-1 + 21) - j(1 - 6) + k(-7 + 2)

a × b = 20i + 5j -5k

Area of Parallelogram = I 20i + 5j -5k I

$Area\:of\:Parallelogram = \sqrt{20^2 + 5^2 + 5^2} = \sqrt{450} = 15 \sqrt{2}$

Answer 2

$<b>$Given that and b are vectors,

Area will be given as ab.

Find ab.

ab= i 1 2
j -1 -7
k 3 1
ab=20i+5j-5k

i.e. √400+50+50

√450

On simplification, 15√2.