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7. Let ㏄ 1 → R, be a cylindrical helix with unit vector u. For to E 1, the curve is called a cro...
5. Let u be a unit vector in R”. Let A = In – uu?. a)....
5. Let u be a unit vector in R”. Let A = In – uu?. a). Verify that A is symmetric, that is, AT = A. b) Verify that A is idempotent, that is, A2 = A. c) Let v be in vector in R”. Show that you can decompose v = w + z where w is a vector orthogonal to u and z is a vector parallel to u. (Hint: Consider the vector projection of v onto u....
component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ...
component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of y and 7 respectively. Let v,v denote the velocity functions of γ and γ respectively. (1) Prove that R 2RV. In particular, at a time t e I for which v(t) lies in the ay-plane,...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
(1 point) Consider the helix r(t)-(cos(-4t), sin (-4t), 4t). Compute, at t A. The unit tangent vector T-( B. The unit normal vector N -( C. The unit binormal vector B( D. The curvature K = Note that...
(1 point) Consider the helix r(t)-(cos(-4t), sin (-4t), 4t). Compute, at t A. The unit tangent vector T-( B. The unit normal vector N -( C. The unit binormal vector B( D. The curvature K = Note that all of your answers should be numbers
(1 point) Consider the helix r(t)-(cos(-4t), sin (-4t), 4t). Compute, at t A. The unit tangent vector T-( B. The unit normal vector N -( C. The unit binormal vector B( D. The curvature K...
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature...
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
Prove the arithmetic properties of the Cross Product 1. 2. a. Line L1 is parallel to the vector u...
please answer question 4-7
Prove the arithmetic properties of the Cross Product 1. 2. a. Line L1 is parallel to the vector u Si+j, line L2 is parallel to the vector u-3i +4j and both lines pass through point P(-1,-2). Determine the parametric equations for line L1 and Lz b. Given line L:x(t)-2t+8,y(t)-10-3t. Does L and Ls has common 3. a. Find the equation of the plane A that pass through point P(3,-2,0) with b. Given A2 be the plane...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu...
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...
5. Let u be a unit vector in R”. Let A = In – uu?. a). Verify that A is symmetric, that is, AT = A. b) Verify that A is idempotent, that is, A2 = A. c) Let v be in vector in R”. Show that you can decompose v = w + z where w is a vector orthogonal to u and z is a vector parallel to u. (Hint: Consider the vector projection of v onto u....
component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of y and 7 respectively. Let v,v denote the velocity functions of γ and γ respectively. (1) Prove that R 2RV. In particular, at a time t e I for which v(t) lies in the ay-plane,...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
(1 point) Consider the helix r(t)-(cos(-4t), sin (-4t), 4t). Compute, at t A. The unit tangent vector T-( B. The unit normal vector N -( C. The unit binormal vector B( D. The curvature K = Note that all of your answers should be numbers
(1 point) Consider the helix r(t)-(cos(-4t), sin (-4t), 4t). Compute, at t A. The unit tangent vector T-( B. The unit normal vector N -( C. The unit binormal vector B( D. The curvature K...
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
please answer question 4-7
Prove the arithmetic properties of the Cross Product 1. 2. a. Line L1 is parallel to the vector u Si+j, line L2 is parallel to the vector u-3i +4j and both lines pass through point P(-1,-2). Determine the parametric equations for line L1 and Lz b. Given line L:x(t)-2t+8,y(t)-10-3t. Does L and Ls has common 3. a. Find the equation of the plane A that pass through point P(3,-2,0) with b. Given A2 be the plane...
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...
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