bn are the columns of b note that x0 c1 b1 c2

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Unformatted text preview: hich of the following quadric surfaces are ellipsoids, which are hyperboloids of one sheet, which are hyperboloids of two sheets, etc.: a. x2 − x2 − x2 − 6x2 x3 = 1. 1 2 3 b. x2 + x2 − x2 − 6x1 x2 = 1. 1 2 3 c. x2 + x2 + x2 + 4x1 x2 + 2x3 = 0. 1 2 3 2.2.8. a. Show that the matrix 1/3 B = −2/3 2/3 2/3 −1/3 −2/3 2/3 2/3 1/3 is an orthogonal matrix of deteminant one. Thus multiplication by B is rotation about some axis through some angle. b. Find a nonzero vector which spans the axis of rotation. c. Determine the angle of rotation. 2.3 Orthonormal bases Recall that if v is an element of R 3 , we can express it with respect to the standard basis as v = ai + bj + ck, where a = v · i, b = v · j, c = v · k. We would like to extend this formula to the case where the standard basis {i, j, k} for R 3 is replaced by a more general “orthonormal basis.” Definition. A collection of n vectors b1 , b2 , . . . , bn in R n is an orthonormal basis for R n if b1 · b1 = 1, b1 · b2 = 0, · · · , b2 · b2 = 1, ·...
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