Question

The points $A(1,2,-1),B(2,5,-2),C(4,4,-3)$ and $D(3,1,-2)$ are

• A. collinear
• B. vertices of a rectangle
• C. vertices of a square
• D. vertices of a rhombus

Answer 1

The correct option is B vertices of a rectangle

AB$=$ $\sqrt{{(2-1)}^2+{(5-2)}^2+{(-2+1)}^2}$
AB$=$ $\sqrt{{\left(1\right)}^2+{\left(3\right)}^2+{(-1)}^2}$
AB$=$ $\sqrt{11}$
Similarly you find that BC$=$ $\sqrt{6}$ CD$=$ $\sqrt{11}$ and DA$=$$\sqrt{6}$
Hence opposite sides of quadrilateral are equal, Now we check the diagonals
AC$=$ $\sqrt{{(4-1)}^2+{(4-2)}^2+{(-3+1)}^2}$
AC$=$ $\sqrt{17}$
similarly BD$=$ $\sqrt{17}$
Diagonals are not equal
direction ratio of line passing through AB is (1,3,-1)
direction ratio of line passing through BC is (2,-1,-1), As the dot product dr of AB and BC are equal to 0 which means AB is perpendicular to BC,similarly check for others sides too
opposite sides are equal and diagonal are equal
hence it is rectangle