Multivariable Caclulus Videos Chapter 9

Given points P and Q in the plane, we find the vector PQ, its magnitude, a unit vector in the direction of PQ and a vector of magnitude 10 in the direction of PQ.

We use a rectangular prism to see why the Pythagorean Theorem extends as it does from 2 dimensions to three.

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We apply the Pythagorean Theorem to the points (2, 5, -4) and (x, y, z), and end up with the equation of a sphere centered at (2, 5, -4).

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We apply the Pythagorean Theorem to obtain the equation of a sphere centered at (x0, y0, z0) having radius r.

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We apply the preceding to obtain the equation of a sphere through (2, 5, -4) with radius 6, and expand the equation into standard form.

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We complete the square to obtain the coordinates of the center, and the radius, of a sphere in standard form.

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We find the points of our sphere which lie directly above and below the point (2, 6, 2).

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We find and sketch the intersection of our sphere with the x-y plane.

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Given three points P, Q and R, we ask whether the vectors PQ, PR and QR lie in the same plane, whether the triangle PQR is a right triangle, and what are the angles between pairs of the three vectors.

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We consider the vector PQ X PR.

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We find the components of a vector F in the direction, and perpendicular to the direction, of another vector `ds. This illustrates the process of scalar and vector projection and two important uses of the projection.

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We apply the cross product to find the area of a parallelogram.

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We now find the volume of a three-dimensional figure whose base is the parallelogram in the previous clip, sides parallel to vector u:

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We now find the volume of a three-dimensional figure whose base is the parallelogram in the previous clip, sides parallel to vector u:

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We develop the equation of a line through the point (x0, y0, z0) and perpendicular to N = a i + b j + c k. We use the process to find specific points through specific planes, and verify important properties of the equation of the plane (e.g., finding a point on a given plane, finding the normal vector given the equation of the plane, consistency among different forms and representations of the plane).

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We find the distance of a point from a plane.

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We find the equation of a line through a given point, parallel to a given vector.

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We find the distance between two lines.

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We find the angle between two planes.

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We consider the general nature of the conic sections formed by the intersections of the quadric surface, given by the equation A x^2 + B y^2 + C z^2 = D, with the coordinate planes.

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We consider the intersection of a specific quadric surface with the xy plane and with another plane parallel to this plane, sketching the resulting conic sections in two dimensions, then within a 3-dimensional coordinate system. We also consider the effect of c on the intersection of the surface with the plane z = c.

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We consider the intersection of the same surface with the y-z plane, and see how the resulting conic section articulates with the sketches of the preceding example.

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We briefly consider the nature and location of the more general quadric surface given by the equation A x^2 +B x + C y^2 + D y + E z^2 + F z + G = 0.

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We consider the intersection of two quadric surfaces.

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The remaining links are not currently occupied. Additional video clips relevant to Chapter 9 may be added in response to questions and/or perceived difficulties.

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