Vector equation of a plane

[Skelly4444]

Please can someone tell me if there's a set procedure for obtaining a vector equation of a plane in the form a = bd1 + cd2, if you're given the Cartesian form of the equation? The example in my further maths book doesn't work when I apply it to a different question. Surely there's a tried and tested method for converting from Cartesian form?

I can successfully do it if it's the other way round and I am converting it into a Cartesian equation.

Thank you in advance.

 

[Dr.Peterson]

Please can someone tell me if there's a set procedure for obtaining a vector equation of a plane in the form a = bd1 + cd2, if you're given the Cartesian form of the equation? The example in my further maths book doesn't work when I apply it to a different question. Surely there's a tried and tested method for converting from Cartesian form?

I can successfully do it if it's the other way round and I am converting it into a Cartesian equation.

Thank you in advance.

Can you give a specific example problem (in each direction), so we can be sure what you mean by a = bd1 + cd2, and show work for what you can do? Which variables are vectors?

 

[Skelly4444]

I meant the form r = A + Bd1 + Cd2 where a is a known point in the plane, B and C are scalar multipliers and d1 and d2 are vectors that are both lie in the plane. It is the form that is presented in most maths books. I just wanted to know how we convert from Cartesian to this form?

 

[blamocur]

I meant the form r = A + Bd1 + Cd2 where a is a known point in the plane, B and C are scalar multipliers and d1 and d2 are vectors that are both lie in the plane. It is the form that is presented in most maths books. I just wanted to know how we convert from Cartesian to this form?

You have a Cartesian form [imath]\mathbf x\cdot \mathbf n = d[/imath] and you need to find [imath]\mathbf, \mathbf v_1, \mathbf v_2[/imath] where [imath]\mathbf p[/imath] is a point on the plane and [imath]\mathbf v_1, \mathbf v_2[/imath] are two linearly independent vectors belonging to the plane.

First find [imath]\mathbf p[/imath] such that [imath]\mathbf p\cdot \mathbf n = d[/imath]. There many solutions to this (the whole plane as a matter of fact ), so you can just use a scaled version of [imath]\mathbf n[/imath], i.e. [math]\mathbf p = \frac{d}{||\mathbf n||^2} \mathbf n[/math]
To find [imath]\mathbf v1, \mathbf v2[/imath] is more difficult. I'd use Gram–Schmidt process by using [imath]\mathbf n[/imath] and a couple of other vectors which are linearly independent with [imath]\mathbf n[/imath]; for example, you can always find 2 out of 3 Cartesian basis vectors which fit the bill.