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Problem 81

Suppose an object moves on the surface of a sphere with |r(t)| constant for all t Show that r(t) and a(t)=r(t) satisfy r(t)a(t)=|v(t)|2

Problem 81

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u,v, and w are vectors in the xy -plane and a and c are scalars. u+v=v+u

Problem 81

Prove or disprove For fixed values of a,b,c, and d, the value of proj (ka,kb)c,d is constant for all nonzero values of k, for a,b0,0.

Problem 81

Relationship between r and r Consider the helix r(t)=cost,sint,t, for \(-\infty

Problem 81

Assume that u,v, and w are vectors in R3 that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2: 1 ratio. The proof does not use a coordinate system. a. Show that u+v+w=0 b. Let M1 be the median vector from the midpoint of u to the opposite vertex. Define M2 and M3 similarly. Using the geometry of vector addition show that M1=u/2+v Find analogous expressions for M2 and M3 c. Let a,b, and c be the vectors from O to the points one-third of the way along M1,M2, and M3, respectively. Show that a=b=c=(uw)/3 d. Conclude that the medians intersect at a point that divides each median in a 2: 1 ratio.

Problem 82

Relationship between r and r Consider the ellipse r(t)=2cost,8sint,0, for 0t2π Find all points on the ellipse at which r and r are orthogonal.

Problem 82

Orthogonal lines Recall that two lines y=mx+b and y=nx+c are orthogonal provided mn=1 (the slopes are negative reciprocals of each other). Prove that the condition mn=1 is equivalent to the orthogonality condition uv=0, where u points in the direction of one line and v points in the direction of the other line..

Problem 82

Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that v×a=κ|v|3B. (Note that T×T=0.) b. Solve the equation in part (a) for κ and conclude that κ=|v×a||v3|, as shown in the text.

Problem 82

Use the formula in Exercise 79 to find the (least) distance between the given point Q and line r. Q(6,6,7),r(t)=3t,3t,4

Problem 82

An object moves along a path given by r(t)=acost+bsint,ccost+dsint, for 0t2π a. What conditions on a,b,c, and d guarantee that the path is a circle? b. What conditions on a,b,c, and d guarantee that the path is an ellipse?

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