Let the π1 plans be: mx − ny + z = 2 and π2: nx − m y + nz =4, with m, n ∈ R. a) Determine m and n so that the planes are parallel. b) Find m and n such that π1 ∩ π2 let be a line perpendicular to the vector u= (2, 1,−1) passing through P = (0, 0, 2)

Q: A rhombus ABCD is such that: 1. A = (-1, 3, -1) 2. B lies on the line I with vector equation: [-1 0…

A:

Q: Use the vector cross product to nd the equation of the plane in R3 containing the three points…

A: Points are  4,1,2,1,-1,1,5,2,3

Q: Consider three points in three-dimensional space: А %3 (15, 2, —3), В %3 (2, —3, 15), С %3 (-3, 15,…

A: we know plane equation is <x-a, y-b, z-c>.n=0

Q: Let U = (2,1,0), V = (1,0,1), and W = (2, −1,1). Find the plane through U, V, and W, expressed in…

A:

Q: Show that the lines L₁ : x = 1 -t, y = 2t, z = 5 + t and L₂: x=1+3t, y = 3-6t, z = 5-3t are parallel…

A:

Q: Let A be the plane containing the points P(2,2,2), Q(2, 1,3 ) and R(3,1, 3) . (a) Find the equation…

A:

Q: 6. Find a plane with the P (3, – 4, 5) and contains the line L:= =*

A: It is given that plane passes through (3,-4,5) and the DR's of a given line is 2,-3,5.

Q: Find an equation of the plane containing the line = 3t, y=2-4t, z = 1+ t and parallel to z-axis.…

A:

Q: 3. Determine the intersection (if any) of the following planes. Show your work clearly ₁: 2x + 3y z…

A:

Q: Find the equ. nf line through the Pont (-2,3,4) and Parallel to Planes: 2X+3y +42=5 3X + 4Y+52=6…

A:

Q: 10.Determine the vector equation of the plane that contains the origin and the intersection of:…

A:

Q: by the equations { L₁ : X Y Let L₁ and L2 be lines in a 3-dimensional space given { = = Z = 3 + 2t…

A:

Q: Prove that the intersection line of the planes P₁: 2x -y - 2z = 0 and P₂:x-2y+z = 0 is parallel to…

A: Since you have posted multiple questions and we can answer one question. So, answered the first…

Q: Consider the points K(−1, 0, 7) and M(1, −2, 5), the vector MN = , and the line L with symmetric…

A: The dot product of two vectors includes the cosine of the angle between them. When the two vectors…

Q: Consider three points in three-dimensional space: A = (11,2, –5), B = (2, –5, 11), C = (-5, 11,2).…

A: Suppose P is a plane passing through a fixed point and a vector perpendicular to the plane or we…

Q: M3. Which of the following expresses the fact that the vectors u and v have the same length? (a) ū -…

A: M3. The two vectors u and v have the same length if: (a)u→·v→=v→·u→

Q: Exercise 6.12 a) Write the cartesian equation of the plane orthogonal too: x-y+2z = 0 and containing…

A: The given plane π is orthogonal to σ : x-y+2z=0 and contain the line  r : x+2y=3x-z=1