Question

How to identify if three vectors lie in a plane or not ?

Answer 1

see if the determinant of the matrix formed by the coordinates of that vector is zero or not........

Answer 2

First of all your vectors (x, y, z) should have the same dimension and if one is null then it does lies in any other plane.
Secondly, you can compare your vectors two by two with the scalar product.
You should remember the formula:
\[x.y=\left| \left| x \right| \right| .\left| \left| y \right| \right|.\cos(x.y)\]
where \[\left| \left| x \right| \right| = \sqrt{x.x}\]and if\[x=\left(\begin{matrix}x_{1} \\ x_{2} \\... \\ x_{n}\end{matrix}\right), y= \left(\begin{matrix}y_{1} \\ y_{2} \\... \\ y_{n}\end{matrix}\right)\]
Then \[x.y = \sum_{i=1}^{n}x_{i}.y_{i}\]
If two vectors lie in the same plane then \[x.y=\left| \left|x\right|\right|.\left|\left|y \right| \right|\] Therefore you can check \[x.y=\left| \left|x\right|\right|.\left|\left|y \right| \right|\] and then that\[x.z=\left| \left|x\right|\right|.\left|\left|z \right| \right|\]
Finding that \[ det(x,y,z) = 0 \] or that \[ \left|\begin{matrix}x_1 & y_1 & z_1\\ x_2 & y_2 & z_3 \\ . & . & . \\ x_n & y_n & z_n\end{matrix}\right| = 0\]will only tell you that at least two vectors amoung the three are colinear (lie in the same plane) but it will not informe you on which one or if the three are all together in the same plane.
Example with \[x=\left( 1,2,3 \right), y =\left( 2,4,6 \right), \left| \left| x \right| \right| = \sqrt{14},\left| \left| y \right| \right|=\sqrt{56}, x.y=28=\sqrt{14}.\sqrt{56}\]
So x and y in that case lie in the same plane

Answer 3

Well, if N vectors line in a 2-dim plane then any linear combination thereof will also lie in that plane. The corresponding matrix will have only that plane as its image (or less than that if all N lie on the same line). So the rank of the matrix will be 2 or less. Therefore, commence Gaussian elimination and see how far you get.

Answer 4

So does these means:
that if I has three vectors: a, b and c, all I need to do is take 2 that is independent of each other in the sense that \[b \neq k'a\] and \[a \neq ka\] where k and k' are some constants. So the third vector, c will be in the same plane as the other if, it is some linear combination of the two i.e. xa + x'b = c. or in matrix form, Ax = c, where A is a matrix with two column a and b respectively, x is a column matrix whose element is x and x', and c is the required vector.
Right?