Question

If $i + j - k$ and $2 i -3 j + k$ are adjacent sides of a parallelogram, then the lengths of its diagonals are

• A. $\sqrt {3},\sqrt {14}$
• B. $\sqrt {13},\sqrt {14}$
• C. $\sqrt {21},\sqrt {3}$
• D. $\sqrt {21},\sqrt {13}$

Answer 1

Answer: (D)

Let $AB = i+ j - k$
and $BC = 2i - 3j + k $

The diagonal $AC = AB + BC$
$=( i + j - k )+( 2 -3 j + k )$
$=3 i -2 j$
The length of diagonal
$AC =\sqrt{(3)^{2}+(-2)^{2}}=\sqrt{9+4}=\sqrt{13}$
The diagonal $DB = AB - AD$
$=( i + j - k )-(2 i -3 j + k ) $
$=- i +4 j -2 k$
The length of diagonal
$DB =\sqrt{1+16+4}=\sqrt{21}$