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Q.
The sides of a parallelogram are $2 i+4 j-5 k$ and $i+2 j+3 k$. The unit vector parallel to one of the diagonals size is
Vector Algebra
Solution:
Let $a=2 i+4 j-5 k, b=i+2 j+3 k$
$\therefore $ diagonals of the parallelogram are
$p=a+b$ and $q=b-a$
i.e., $p=3 i+6 j-2 k$
and, $q=-i-2 j+8 k$
$\therefore $ unit vectors along the diagonals are
$\frac{3 i+6 j-2 k}{\sqrt{9+36+4}}$ and $\frac{-i-2 j+8 k}{\sqrt{1+4+64}}$
i.e., $\frac{3 i+6 j-2 k}{7}$ and $\frac{-i-2 j+8 k}{\sqrt{69}}$
$\therefore $ unit vector parallel to one of the diagonals is
$\frac{1}{7}(3 i+6 j-2 k)$