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

Question

Consider the planes 3x+4y+4z=1 and 3x+4z=0 (A) Find the unique point P on the y-axis which is on both planes: (0, ¼, 0)(B)Find a unit vector u with positive first coordinate that is parallel to both planes.  i + __j + k(C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r(t)= i + __j + k

Mr. Toni Zulauf Sr. · Accepted Answer

A) point on the y axis and on both planes:on the y axis means: x and z = 0so set x and z = 0 in both planes, and y must be equal in both planes3x + 4y + 4z = 1 ---> 4y = 13x + 4z = 00 = 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) = 20unit 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)kOG

Mr. Toni Zulauf Sr. · 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

Sigrid O'Kon

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

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