Which of these planes are perpendicular to lines r=(0,1,1) + t(1,1,0)?
a) x+y+z=2
b) x+y+z=3
c) y+z=2
d) x+y=2
Which of these planes are perpendicular to lines r=(0,1,1) + t(1,1,0)? a) x+y+z=2 b) x+y+z=3 c)...
Which of the following vectors are perpendicular to the lines r=<0,1,1> + t<1,1,0> and r=<1,0,0>+t<0,1,1>? a) <0,0,1> b) <-1,0,0> c) <1,-1,1> d) <0,1,0>
1. Given force F-xi 4zj Path 1: (0,0,0) to (0,1,1) Path 2: (0,1,1) to (1,1,0) Path 3: (1,1,0) to (0,0,0) 5yk acts following the path: . Sketch the motion in Cartesian coordinate Write the parametric equation for each path Calculate the total work done by force F which is moving from Path 1 to 3 2. A vector M is defined by M-x2yi xyj and region R corresponds to Sketch the region of R in Cartesian coordinate . Evaluate the...
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t) =t+4 12: x(u) = 5 - 3u, y(u) = u +1, (u) = u +2 1. Determine whether li and ly are parallel, skew or intersect. If the lines intersect, find the point of intersection of li and 12. 2. If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the lines are skew, find a...
1. (Sections 2.11,2.12) The parametric equations of three lines are given below: 4 : (x, y, z) = (1,0,0) + (1,0,-1), TER 19 : (x,y, 2) = (1.0.-1) + (0.1, -1), TER 13: (1.7.2) = (1.-1,-1) + (1,1,0), TER Two of these lines intersect. Which two? What is the equation of the plane that they describe? Give full reasons for your answers. 2. (Sections 2.11,2.12) Given the two planes 2-y-z = 0 and r+y-:-1=0. Find a parametric equation for the...
x =-y+2 = -z+2 The symmetric equations for 2 lines in 3-D space are given as: 1. L,: x-2 = -y+1 = z+1 a) Show that lines L1 and L2 are skew lines. b) Find the distance between these 2 lines x =1-t y=-3+2t passes through the plane x+ y+z-4=0 2. The line Determine the position of the penetration point. a. Find the angle that the line forms with the plane normal vector n. This angle is also known as...
log(2 - 2) Consider the function f(x, y,z) (a) What is the maximal domain off? (Write your answer in set notation.) Find ▽f. (b) Find the tangent hyperplanes Ta2.1,f(r, y, 2) and To-ef(r, y, 2). Find the intersection (c) On (z, y, z)-axes, draw arrows representing the vector field F = Vf at the points (1,0,1), (d) Find the level set of f which has value ("height") wo 0, and describe it in words and of these two hyperplanes, and...
Please help with these problems. 8. Consider the two planes listed below 2x - y + z = 1 +y-2=2 These two planes intersect at a right angle. Show that this is true by showing their normal vectors are perpendicular. Find the parametric equations of their line of intersection. Is the line of intersection (call this L) for these planes parallel, perpendicular (intersect at 90 degrees), skew (not parallel, don't intersect), or none of the above to the line: F(t)...
Which of the following defines an inner product on R^3 <(x,y,z),(a,b,c)>= xa+2xb+3xc <(x,y,z),(a,b,c)>= xy+za+bc <(x,y,z),(a,b,c)>= xa-yb+zC <(x,y,z),(a,b,c)>= (x+z)(a+c)+(2x+2y)(2a+2b)+(3x+z)(3a+c)
2. [& marks] Consider the line ar transformation T: R – R? T(x,y,z) = (x +y-2, -1-y+z). (a) Show that the matrix [T]s, representing T in the standard bases of Rand R' is of the form [7|6,6= ( +1 -1 1). -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d)...
X-2 Gi y + 1 -2 and x=t, y = 1 + t, and z=1-t, a) find the distance between the origin and the first line, b) find the angle between the two lines, and c) find an equation for the plane which contains both lines.