Calculating Perpendicular Point on a Line in 2D Plane

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To calculate the point C that makes AC perpendicular to BC on a line L in a 2D plane, the relationship involves substituting C into the equation and ensuring the inner product <AC, BC> equals zero. The vectors AC and BC can be expressed in terms of the line's parameters and points A and B. The inner product simplifies to an expression involving the dot products of these vectors, but without specific values for A, B, v0, and n, further simplification is challenging. The discussion highlights the complexity of deriving a more straightforward solution without concrete data. Ultimately, the mathematical relationship remains intricate and requires specific inputs for resolution.
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Hi all:

Given a line L:v= v0+t*n; and two points A, B in 2D plane; A and B are on the two sides of the line L. I want to calculate the point C which makes AC is perpendicular to BC

I know it's simply that substitude v to C and <AC, BC>=0. But I don't know how to simplify the equation.

Could anyone help me please?

Thanks
 
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Asuralm said:
Hi all:

Given a line L:v= v0+t*n; and two points A, B in 2D plane; A and B are on the two sides of the line L. I want to calculate the point C which makes AC is perpendicular to BC
C is on L?

I know it's simply that substitude v to C and <AC, BC>=0. But I don't know how to simplify the equation.

Could anyone help me please?

Thanks
The vector AC= v0+ t*n is given by v0+ t*n-A. The vector BC is given by v0+ t*n- B.
Their inner product <AC,BC>= <v0+ t*n- A,v0+ t*n- B>= |v0- t*n|2- <v0+ t*n,A+B>+ <A,B>.
Without specific values for A and B, v0 and n, I don't see how you can get any simpler than that.
 
should this |v0- t*n|2- <v0+ t*n,A+B>+ <A,B>

be <v0+t*n, v0+t*n> - <v0+t*n, A+B> + <A, B> ?
 

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