Question

Find the area of the parallelogram whose diagonals are : 4i - j - 3k and -2i + j - 2k

Answer 1

Given two diagonals of a parallelogram are \(4\hat i-\hat j-3\hat k\) and  \(-2\hat i+\hat j-2\hat k\)

Let  \(\vec a= 4\hat i-\hat j-3\hat k\) and  \(\vec b=-2\hat i+\hat j-2\hat k\)

Recall the area of the parallelogram whose diagonals are given by the two vectors \(\vec a=a_1\hat i+a_2\hat j+a_3\hat k\)  and  \(\vec b = b_1\hat i+b_2\hat j+b_3\hat k\) is  \(\cfrac12|\vec a\times\vec b|\) where

Here, we have (a₁, a₂, a₃) = (4, –1, –3) and (b₁, b₂, b₃) = (–2, 1, –2)

Recall the magnitude of the vector  \(\text x\hat i+y\hat i+z\hat k\)  is

Thus, the area of the parallelogram is 7.5 square units.