(a) Show graphically that line plane relational {x|axx=d} {z | Az = D} parametric {* =...

 

Transcribed Image Text:

(a) Show graphically that line plane relational {x|axx=d} {z | Az = D} parametric {* = ₁ + aa | a ER} { t = ₂ + b3 + cy | B, YER} define a set of points on a line or plane, where a, d, A, D are constants which define the geometry, 71,2 are fixed points on the line and plane respectively, and a, 3, y are parameters that vary along the line/plane (they uniquely parametrize points in the line/plane). [bonus:] Show the 5th relation { = ₂ + Ax8| 8 € R³}. Bonus: Show the fifth relation { = ₂ + Ã × ♂ | ♂ € R³}. (b) What constraint between a and d is implicit in the formula a x = d? Likewise, what relations between b, c, and à must be satisfied if both equations describe the same plane? For the line and the plane, substitute from the parametric form into the relational form to show that they are consistent. What are d and D in terms of a, ₁ and Ã, T2, respectively? (a) Show graphically that line plane relational {x|axx=d} {z | Az = D} parametric {* = ₁ + aa | a ER} { t = ₂ + b3 + cy | B, YER} define a set of points on a line or plane, where a, d, A, D are constants which define the geometry, 71,2 are fixed points on the line and plane respectively, and a, 3, y are parameters that vary along the line/plane (they uniquely parametrize points in the line/plane). [bonus:] Show the 5th relation { = ₂ + Ax8| 8 € R³}. Bonus: Show the fifth relation { = ₂ + Ã × ♂ | ♂ € R³}. (b) What constraint between a and d is implicit in the formula a x = d? Likewise, what relations between b, c, and à must be satisfied if both equations describe the same plane? For the line and the plane, substitute from the parametric form into the relational form to show that they are consistent. What are d and D in terms of a, ₁ and Ã, T2, respectively?