A First Course in Linear Algebra by Daniel Zelinsky and Samuel S. Saslaw (Auth.)

By Daniel Zelinsky and Samuel S. Saslaw (Auth.)

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Strategy: If we can find a point Po on the plane and a vector ν perpendicular to the plane, then the distance from the origin 0 to the plane is the absolute value of II OPo II cos Ö where θ is the angle between OPo and ν (see Fig. 1). Since OPo · ν = || OPo || || ν || coso, the answer to our question is I OPo · V I SubproUem 1. Answer: 2i + 3j + 4k. Subprobkm 2. ) Find a point Po on the plane. This is only 1. Planes 59 difficult because there are so many correct answers. For ex­ ample, choose any y and ζ and find χ to match.

D) V · (w + w' + w") = v w + v w ' (e) (v + v' + v'O · w = so on. V + v w". · w + v' · w + v'' · w, and Note that the associative law ( u · v) · w = u · (v · w) cannot hold. It does not even make sense since u · ν is a scalar and you cannot take a dot product of a scalar and a third vector. It is also false in general that ( u · v ) w = u ( v · w) where on each side of the equation you have one dot product and one multiphcation of a vector by a scalar; the left side is a vector parallel to w and the right side is a vector parallel to u , so the two cannot be equal unless u and w are parallel.

8. 12. 9 thus: (u X v) · w = (v X w) · = — (v X u) = X — (u = (w u X u) · w = — (w w) · · V X v) · u V. 8. Vectors in n-Space The analytic aspect of vectors with tail at the origin—they are triples of numbers—is clearly meat for generahzation. We say that a vector in n-space is an n-tuple of numbers. We denote by R"" the set of all such vectors in n-space (Ä stands for real numbers). If η = 1, 2, or 3 we can associate with a vector in a geometric object: an arrow on the line, in the plane, or in 3-space, respectively (always with tail at the origin).

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