calc 3 – Consider the planes 3x+4y+4z=1 and 3x+4z=0?

Question

Consider the planes defined by the equations (3x + 4y + 4z = 1) and (3x + 4z = 0).
(A) Find the unique point (P) on the (y)-axis that lies on both planes. 
(B) Determine a unit vector (u) with a positive first coordinate that is parallel to both planes, expressed in the form (i + __j + k).
(C) Using the results from parts (A) and (B), derive a vector equation for the line of intersection of the two planes, represented as (r(t) = i + __j + k).

Anonymous · Accepted Answer

A) 
point on the y axis and on both planes:
on the y axis means: x and z = 0
so set x and z = 0 in both planes, and y must be equal in both planes
3x + 4y + 4z = 1 ---> 
4y = 1
3x + 4z = 0
0 = 0

y = 1/4 
point is (0, 1/4; 0) 
B)
A vector parallel to both planes is the cross product vector from both normal vectors.
vector = (3, 4, 4)x(3, 0, 4) = (16-0, 12-12, 0-12) = (16, 0, -12)
A unit vector is a vector divided by its magnitude: magnitude = √(16^2+12^2) = 20
unit vector = ( 16/20, 0, -12/20) or (4/5, 0, - 3/5) 
C) 
An equation for the intersection line is 
line: (x, y, z) = (0, 1/4, 0) + t(4/5, 0, - 3/5)
or 
(4/5*t)i + (1/4)j + (- 3/5t)k
OG

Tamia Moen · Answer

2y + x = 3 x = 3 - 2y change x in 2d equation: 4y - 3x = a million 4y - 3(3 - 2y) = a million 4y - 9 + 6y = a million 10y - 9 = a million 10y = 10 y = a million Plug y back in first equation: x = 3 - 2y x = 3 - 2(a million) x = 3 - 2 x = a million

Anonymous

calc 3 – Consider the planes 3x+4y+4z=1 and 3x+4z=0?

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