Show that except in degen-erate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w) lies in the plane of v and w. What are the degenerate cases?

Q: The region bounded on the left by the y-axis, on the right by the hyperbola x2 - y2 = 1, and above a...

A: Given,The region bounded on the left by the y-axis, on the right by the hyperbola

Q: Parametrize the line that passes through the point (1, 2, 3) parallel to the vector v =-3i + 7k.

A: WE know that equation of line through (x1,y1,z1) and parallel to vector ai+bj+ck is 

Q: Find the centroid of the region enclosed by the x-axis and the cycloid arch x = a(t - sin t), y = a(...

A: Given data:

Q: Give some typical parametrizations for lines, circles, parabolas, ellipses, and hyperbolas. How migh...

A: First of all, consider the cartesian form of the equation for the line.Then, consider the parametriz...

Q: Show that a unit vector in the plane can be expressed as u = (cos u)i + (sin u)j, obtained by rotati...

A: To show that a unit vector can be expressed as u = (cos u) i + (sin u) j.

Q: What is an alternating series? What theorem is available for deter-mining the convergence of such a ...

A: The alternating series test.

Q: Most centroid calculations for curves are done with a calculator or computer that has an integral ev...

A: Click to see the answer

Q: Sketch the surfaces 1. . z = x2 + 4y2  2. x = 4 - 4y2 - z2 3. x2 + y2 = z2

A: The given surfaces are sketched using online 3D calculator.

Q: Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the re...

A: Sketch the integrant as shown below.