5. Let P be the unique plane in R3 containing the points (1,3,2), (1,-1,0), and (2,0,1)....
5,6 please 5. Parametrize the plane P in R3 containing the points x (1,0, ), x (2,0, 1) and x3 (1,3, 1). Does the plane P contain the point (-1,3,2)? 6. Sketch and parametrize the triangle in the plane with vertices x! = (-1,-2), x2 = (2, and x3 (1,3). Does this triangle contain the origin (0, 0,0)
Problem 1. Let P be the plane in R3 with parametric equation and let span | | 21 , 10 (Note that Q is a plane containing the origin.) Determine the intersection of P and Q Problem 1. Let P be the plane in R3 with parametric equation and let span | | 21 , 10 (Note that Q is a plane containing the origin.) Determine the intersection of P and Q
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Consider the points: P (-1,0, -1), Q (0,1,1), and R(-1,-1,0). 1.) Compute PQ and PR. 2.) Using the vectors computed above, find the equation of the plane containing the points P, Q, and R. Write it in standard form. 3.) Find the angle between the plane you just computed, and the plane given by: 2+y+z=122 Leave your answer in the form of an inverse trigonometric function.
Question 1 2 pts Describe the span of {(1,0,0),(0,0,1)} in R3 The x-z plane R3 R2 The x-y plane Question 2 2 pts Describe the span of {(1,1,1),(-1,-1, -1), (2,2, 2)} in R3 A plane passing through the origin Aline passing through the origin R3 A plane not passing through the origin A line not passing through the origin Question 3 2 pts Let u and v be vectors in R™ Then U-v=v.u True False Question 4 2 pts Ifu.v...
5. (10pts) Let B (v1 (1,1,0), v2 (1,0,-1). v3 (0,1,-1)) be a basis of R3 Using the Gram-Schmidt process, find an orthogonal basis of R3. (You don't have to normalize the vectors.)
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
1. (15 points) (a) (5 points) Find the equation of the plane a that contains points A(1,5,4) B(1,0, 1) and C(4, 0,5) (b) (5 points) Find the distance from the point D(2, 1,7) to this plane (c) (5 points) If plane 3 has equation y -3z+2x = 5, find a unit vector that is parallel to the intersection of a and B.
6. (20 points) Let W be a plane spanned by the vectors ői = [1, 2, 2)", T2 = (-1,1,2) (a). Find an orthonormal basis for W. (b). Extend it to an orthonormal basis of R3.