Question

The points $(5,-4,2),(4,-3,1),(7-6,4)$ and $(8,-7,5)$ are the vertices of

• A. A rectangle
• B. A square
• C. A parallelogram
• D. None of these

Answer 1

Answer: (C)

A parallelogram

Let $A=(5,-4,2),B=(4,-3,1),C=(7,-6,4)$ and $D=(8,-7,5)$.
Now, $AB=\overline{4-5^2+-3+4^2+1-2^2}$
$\overline{1+1+1}=\overline{3}$
$BC=\overline{8-7^2++-7+6^2++5-4^2}$
$\overline{1+1+1}=\overline{3}$
$CD=\overline{8-7^2++-7+6^2++5-4^2}$
$\overline{1+1+1}=\overline{3}$
And $AD=\overline{8-5^2+-7+4^2+5-2^2}$
$\overline{\overline{9+9+9}}=3\overline{3}$
Again Now, position vectors of
$AB=(4-5)i+(-3+4)j+(1-2)k$
$=-i+k-k$
$BC=(7-4)i+(-6+3)j+(4-1)k$
$=3i-3j+3k$
Therefore,$AB.BC=(-1+j-k).(3i-3j+3k)$
$=-333\ne 0$
Therefore, $ABCD$ is a parallelogram.