Question

Points $(-2,4,7)$, $(3,-6,-8)$ and $(1,-2,-2)$ are:

• A. Collinear
• B. Vertices of an equilateral triangle
• C. Vertices of an isosceles triangle
• D. None of the above

Answer 1

The correct option is A

Collinear

Explanation for the correct option:

Formula to be used : We know that three points $\left(x_1,y_1,z_1\right)$, $\left(x_2,y_2,z_2\right)$ and $\left(x_3,y_3,z_3\right)$ are collinear if the value of the determinant

$\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|\mathbf{=}\mathbf{0}$

Here, given points are : $(-2,4,7)$, $(3,-6,-8)$ and $(1,-2,-2)$.

Then, using the above formula, the required determinant will be

$$\begin{gathered} \left|\begin{array}{ccc}-2& 4& 7\\ 3& -6& -8\\ 1& -2& -2\end{array}\right| \\ =-2\left[\left(-6\right)\times \left(-2\right)-\left(-2\right)\times \left(-8\right)\right]-4\left[3\times \left(-2\right)-1\times \left(-8\right)\right]+7\left[3\times \left(-2\right)-1\times \left(-6\right)\right] \\ =-2\left(12-16\right)-4\left(-6+8\right)+7\left(-6+6\right) \\ =8-8+0 \\ =0 \end{gathered}$$

Since the determinant is Zero, the given points $(-2,4,7)$, $(3,-6,-8)$ and $(1,-2,-2)$ are collinear.

Hence, the correct answer is option (A).