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(a) Find and identify the traces of the quadric surface $ -x^2 - y^2 + z^2 = 1 $ and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1.(b) If the equation in part (a) is changed to $ x^2 - y^2 - z^2 = 1 $, what happens to the graph? Sketch the new graph.

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03:35

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 6

Cylinders and Quadric Surfaces

Vectors

Johns Hopkins University

Oregon State University

Harvey Mudd College

Baylor University

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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$\begin{array}{l}{\text { …

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(a) Find and identify the …

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Graph the quadric surfaces…

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01:40

So here we want to find and identify the traces of the quadratic surface or aquatic surface. So it's going to be negative acts squared minus y squared minus z squared one. Um And we want to explain why the graph looks like the graph of a hyperba Lloyd of two sheets. And that's because the traces parallel to the y z plane are going to be hyperbole as and if we change the equation uh so this is actually plus one R plus t squared. And if we change it to minus the squared, then we end up getting that the traces parallel to xz the xz plane will be hyper bolos. Um So that would be the resulting graph that we end up getting. And it's going to be able to form Z squared minus x squared equals one plus k squared or z squared minus y squared equals one plus k squared.

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