Question

If the plane passes through three collinear points \((x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)\) then which of the following is true?

(a) \(x_1 y_1 z_1+x_2 y_2 z_2+x_3 y_3 z_3\)=0

(b) \(\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}\)=0

(c) \(\begin{vmatrix}x_1\\y_2\\z_3\end{vmatrix}\)=0

(d) \(x_1 x_2 x_3+y_1 y_2 y_3+z_1 z_2 z_3=0\)

The question was asked in an interview for internship.

My question comes from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Right answer is (b) \(\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}\)=0

The explanation: If the three points are collinear, then they will be in a straight line and hence the determinant of the three points will be zero.

i.e.\(\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}\)=0