Lines And Planein 3D

Question 1

What are properties of STP

Answer 1

Properties of STP are:
[a,b,c]=[b,c,a]=[c,a,b] (cyclic)
= -[a,c,b] = -[b,a,c] = -[c,b,a]
[a,b,a]=0
[a,b,c+d]= [a,b,c] + [a,b,d]
L[a,b,c] = [La,b,c]

Question 2

What are the 2 forms of the triple vector product

Answer 2

2 forms of the triple vector product are:
A x (b x c)= (c.a)b - (b.a)c
(A x b) x c = (a.c)b - (b.c)a

Question 3

What is the Jacobi identity

Answer 3

Jacobi identity is:
A x (b x c) + b x (c x a) + c x (a x b) =0

Question 4

What is the vector equation for a straight line

Answer 4

Vector equation for a straight line is:

R = a + Ul where a is a point in the plane and l is a direction vector, U is a scalar.

Question 5

What is the alternative form of the equation of a straight line

Answer 5

Alternative form of the equation of a straight line is:
AR is parallel to l iff
(r-a) x l =0

Question 6

What is the equation of a plane

Answer 6

Equation of a plane is:

Ax+by+cz+d=0

Question 7

What is the parametric equation of a plane

Answer 7

Parametric equation of a plane is:

R = a + lambda(L) + mu(M) where L and M are direction vectors

Question 8

What is implicit equation of a plane

Answer 8

Implicit equation of a plane is:
(R-a).N =0 where N is perpendicular to l x m.
R.N=A.N (constant)

Question 9

What is the unit normal

Answer 9

Unit normal is n= +- (l x m)/norm(l x m)

Question 10

What is the geometric meaning of d in r.N=d equation

Answer 10

Geometric meaning of d in r.n=d (where n is unit normal) equation is:
D= +-norm(r) where r is a position vector in the plane such that OR is normal to the plane. Perpendicular distance from O to plane.