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(Fall15 exam 1 problem 1 modified) A line segment connects the points (1,2,3) and (2,4,4). (a)...
Carefully draw the line segment PQ that connects P=(4, 5, -3) and Q=(0, -4, 2) . Include dotted vertical lines from the xy-plane to P and Q to show perspective. Find the distance between P and Q, from the previous problem. Then find the coordinates of the midpoint of the line segment PQ . Let u= -3i+5j+7k and v= 10i+j-2k . Show that u × v is orthogonal to the vector v .
Question 1 (1 point) A line segment AD, contains points B & C such that C is between A and D, and B is between A and C. If AB- 6, BD- 23, and AB -CD, find the length of segment BC. 17 23 29 Not enough information to solve the problem
3. (5 points) (a): Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=etcost, yr etsint, z=et; (1,0,1) (b): Find the unit tangent vector T, the principal unit normal N, and the curvature k for the space curve, r(t) =< 3 sint, 3 cost, 4t >.
6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3) is the vector (2, 1, 1) and g(1,2,3) = 1 (1) (2 points) Find the equation of the tangent plane of the level surface g(r, y, z) = 1 at (1,2,3) (ii) (2 points) Find the maximum rate of change of g(x, y, z) at (1, 2, 3). hax. rarte ot change: 23 14 (iii) (2 points) Find the rate of change of g...
Let the vectors a = <1,2,3>, b= <1,1, 1 > and c = <1,2, 1 > a) Determine whether the three are coplanar None of these 4 0.71 0.74 no b) Find the volume of the parallelepiped form c) Find the unit vector orthogonal to both ved d) Find the angle between the vectors a and 22.26 bunded to 2 decimal points) 12:21 e) Find the component of the vector a along 39.51 -1 ge
8. Find the length of the curve given by F(t)=(3 sin(21),19,3cos(21), for ISIS 3, rounded to the nearest tenth. (6 points) 9. Suppose that a space curve given by the vector function () = (21'.1'. 36). a. Find parametric equations for the tangent line to this space curve at the point where - (4 points) b. Find the unit tangent vector, the unit normal vector, the unit binormal vector and the curvature for this space curve at the point where...
Chain Rule Tangent Planes: Problem 10 Previous Problem Problem List Next Problem (1 point) Consider the surface xyz = 6. A Find the unit normal vector to the surface at the point (1,2,3) with positive first coordinate. 0 B. Find the equation of the tangent plane to the surface at the given point Express your answer in the form ar+by+c+d normalized so that a 6. Note: You can earn partial credit on this problem Submit Answers Preview My Answers You...
Line Segment? -- + + ok -Pas-pa) .) to c leng of jlt) = <-sin (36), is 710) = 3i +25 +6K t (st?+1)4 given that the velocity is the initial position esty od Section 13.1 Parameterization 1. Uw the start point the point he cotire segment as the din 9.) Find the position fonction Adu PO Line Segment Section 12.2 Vector Component Form: - (s. 12.) or + 1) + wyk actor from PP. Papa) to .92.a): PO -...
Given the points
,
and
:
a) Find the midpoint M of the segment
and draw it.
b) Find the length of the vector
.
c) Find the angle between
and
.
Let A = (-1, -2,0), B = (1,0,1), and C = (-1,2,3) be points in space. (a) Find the vector projection of BA onto the vector BC. (b) Let @ be the angle between BÀ and BC. Compute cos(20).