By S. A. Amitsur, D. J. Saltman, George B. Seligman

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The matrix product may be written as a linear combination of the column vectors a and b ¸ ¸ ¸ u 0 d1 e1 = v 0 d2 e2 ¸ ¸ d1 u + e1 v 0 = 0 d2 u + e2 v ¸ ¸ ¸ e1 0 d1 +v = u 0 d2 e2 ua + vb = 0= Thus, the above are also equivalent to ua + vb = 0 implies both u and v must be zero. 4. 1. 3 the area of the larger parallelogram, formed by the scaled vectors ua and vb> is uv times the area of the smaller parallelogram. 3 Applications to Work and Torque This application is concerned with moving a mass from one point in a plane to an other point in the plane.

If the displacement vector is g = [30 6]> ﬁnd the work. $ $ 14. Use MATLAB to compute the dot product of d = [2 1] and e = [3 4]= $ d = 15. 5 Lines and Curves in R2 and C In this section lines in the plane will be represented by three variations on algebraic equations and three variations on vector equations. We will use these descriptions to ﬁnd the point on a line that is closest to a given point not on the line. One method uses calculus to solve the problem, and the other method uses geometry to generate the same solution.

In fact, this can be simpliﬁed to easily compute cos()> sin() and the area of the parallelogram formed by two vectors. 4 this will be applied to the computation of work and torque. 3. 4. If d = [2 4] and e = [1 2]> then $ d e = [3 2]> ° ° ° ° 2 2 $° $° ° ° 2 $ d k = 20> ° e ° = 5 so that d e ° = 13> k °$ s s 20 + 5 2 20 5 cos() 20 + 5 20 cos() 12@20 = 3@5 cos1 (3@5) 53=13 degrees. 3 Application to Navigation of an Aircraft A pilot wishes to ﬂy northeast with a speed of 400 miles per hour.