# determinant can represent a plane?

• Feb 2nd 2013, 07:32 PM
happymatthematics
determinant can represent a plane?
Attachment 26824
Why the above determinant can represent a plane?
Can anyone interpret it?
Thank you.
• Feb 2nd 2013, 09:37 PM
chiro
Re: determinant can represent a plane?
Hey happymatthematics.

Hint: Try expanding it and relate the cross product of vectors to the plane normal.
• Feb 3rd 2013, 10:27 PM
hollywood
Re: determinant can represent a plane?
You ask only if it represents a plane, but I think you can say more. It actually represents the plane through $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, and $(x_3,y_3,z_3)$, doesn't it?

- Hollywood
• Feb 8th 2013, 07:19 AM
happymatthematics
Re: determinant can represent a plane?
Hello chiro and hollywood,
Yes I realized that it represents a plane passing through (x1,y1,z1) , (x2, y2, z2) and (x3, y3, z3).
but is it a must that the last number in each row is 1 in this determinant?
• Feb 8th 2013, 10:08 AM
johng
Re: determinant can represent a plane?
Yes, the last column must be all the same non-zero value if you want the plane to pass through each of the points (x1,y1,z1), etc. So you might as well make this common value 1.

To see why you get a plane, expand the determinant along the first row and you get ax +by +cz + d =0, where a, b, c and d are cofactors. Now to see (x1,y1,z1) satisfies this equation, replace the first row x y z 1 by x1 y1 z1 1, and you have a determinant with two rows the same, so the determinant is 0. Notice though if the three points are collinear, you don't get a plane at all; the determinant is identically 0!
• Feb 9th 2013, 10:42 PM
happymatthematics
Re: determinant can represent a plane?
Thank you. Then How to interpret This three point form equation of a plane when all the 1 on the rightmost is converted to 2?
• Feb 10th 2013, 09:52 PM
hollywood
Re: determinant can represent a plane?
Multiplying a column by 2 multiplies the determinant by 2, so when you set it equal to 0 and cancel 2, you end up with the same equation.

- Hollywood