Question

The points P,Q,R have rectangular cartesian coordinates(1,1,-1),(4,1,2),(-2,1,2)respectively. Which of the following is true for the triangle PQR ?

• A. All the sides have the same length
• B. QR=2 PQ
• C. Angle PQR is a right angle
• D. The area of the triangle is 18 square units
• E. All statements are false

Answer 1

Answer: (E)

All statements are false

For the above points to form a triangle or a 2D figure, they must lie on the same plane.

Now

Let

$P=(1,1,-1)$

$Q=(4,1,2)$

$R=(-2,1,2)$

Then

$PQ=\overline{a}=3i+0j+3k$

$|a|=3\sqrt{2}$

$QR=\overline{b}=-6i+0j+0k$

$|b|=6$

$RP=\overline{c}=3i+0j-3k$

$|c|=3\sqrt{2}$

Now

$(\overline{a}\times \overline{b})=-12j$

Now

$\overline{c}.(\overline{a}\times \overline{b})$

$=(3i-3k).(-12j)$

$=0$

Thus they are co planar.

Also

$|a{|}^2+|c{|}^2=|b{|}^2$

Also

$PQ$ is perpendicular to $RP$

Hence the PQR is an isosceles right angled triangle right angled at P with

$PQ=PR$.