Question

Prove that the points $(4,5,-5),(0,-11,3)$ and $(2,-3,-1)$ are collinear.

Answer 1

Method-1:

$\left|\begin{array}{ccc}4& 5& -5\\ 0& -11& 3\\ 2& -3& -1\end{array}\right|=4(11+9)+2(15-55)=80-80=0$

$\therefore$ the three points are collinear.

Method-2:

Distance formula in 3D:

Distance between two points $(x_1,y_1,x_1)and(x_2,y_2,z_2)=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Let us call the points as $A(4,5,-5)$, $B(0,-11,3)$ and $C(2,-3,-1)$.

Now, we can calculate the distances $\overrightarrow{AB}$, $\overrightarrow{BC}$ and $\overrightarrow{AC}$.

$\overrightarrow{AB}=\sqrt{4^2+{16}^2+8^2}=\sqrt{336}=4\sqrt{21}$

$\overrightarrow{BC}=\sqrt{2^2+8^2+4^2}=\sqrt{84}=2\sqrt{21}$

$\overrightarrow{AC}=\sqrt{2^2+8^2+4^2}=\sqrt{84}=2\sqrt{21}$

$\overrightarrow{BC}+\overrightarrow{AC}=\overrightarrow{AB}$.

$\therefore$ $A,B$ and $C$ lie on a straight line.

Thus, the points are proved to be collinear.