Elementary Linear Algebra by Stephen Andrilli; David Hecker

By Stephen Andrilli; David Hecker

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Vectors are added by summing their respective coordinates. For example, if x ϭ [2, Ϫ3, 5] and y ϭ [Ϫ6, 4, Ϫ2], then x ϩ y ϭ [2 Ϫ 6, Ϫ3 ϩ 4, 5 Ϫ 2] ϭ [Ϫ4, 1, 3]. Vectors cannot be added unless they have the same number of coordinates. There is a natural geometric interpretation for the sum of vectors in a plane or in space. The total movement (x ϩ y) is equivalent to first moving along x and then along y. 7 illustrates this in R2 . Let Ϫy denote the scalar multiple Ϫ1y. We can now define subtraction of vectors in a natural way: if x and y are both vectors in Rn , let x Ϫ y be the vector x ϩ (Ϫy).

Using Newton’s Second Law of Motion, find the acceleration vector on a 6-kg object in a three-dimensional coordinate system when the following two forces are simultaneously applied: f1 : A force of 22 newtons in the direction of the vector [9, 6, Ϫ2] f2 : A force of 27 newtons in the direction of the vector [7, Ϫ4, 4] 20. Using Newton’s Second Law of Motion, find the resultant sum of the forces on a 30-kg object in a three-dimensional coordinate system undergoing an acceleration of 6 m/sec2 in the direction of the vector [Ϫ2, 3, 1].

For example, if t is a real number, then t > 5 ⇒ t 2 > 25 is a valid step, but reversing this yields t 2 > 25 ⇒ t > 5, which is certainly an invalid step if t < Ϫ5. Notice that we were very careful in Step 8 of the proof when we took the square root of both sides to ensure the step was indeed valid. “If A Then B” Proofs Frequently, a theorem is given in the form “If A then B,” where A and B represent statements. ” The entire“If A then B”statement is called an implication; A alone is the premise, and B is the conclusion.

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