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Throwback
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Homework Statement
Given the following 2-dimensional plane in R^4 written in parametric form:
x_1 = 1 + (-1)s
x_2 = 0 + ( 1)s
x_3 = 1 + (-1)t
x_4 = 0 + ( 1)t
a) find a 2-dimensional plane that does not intersect it
b) find a 2-dimensional plane that intersects it to form a point
c) find a 2-dimensional plane that intersects it to form a line
Homework Equations
dot product
The Attempt at a Solution
a) Just change the position vector, so what results is a parallel line, shifted away from the original plane, and therefore cannot intersect:
x_1 = 0 + (-1)s + (0)t
x_2 = 0 + (1)s + (0)t
x_3 = 0 + (0)s + (-1)t
x_4 = 0 + (0)s + (1)t
b) The dot product between the original plane and my new plane must be equal to zero, and hence they are perpendicular, forming a point. I'm not quite sure how to calculate this... Is this even possible (i.e., trick question)? Help!
c) I need the same position vector (1,0,1,0), but what about the direction vectors? I guess they shouldn't be perpendicular here, so... I'm really not sure... What if I change just one number in one of the direction vectors?
x_1 = 1 + (-1)s + (0)t
x_2 = 0 + (0)s + (0)t
x_3 = 1 + (0)s + (-1)t
x_4 = 0 + (0)s + (1)t
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