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Lecture 1 - Vectors.pdf

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MAC 2313
Jason Kozinski

Lecture 1 (Text Sections 12.1 and 12.2) Vectors A vector is a directed line segment and it is represented graphically as an arrow in space with the direction of the vec- tor given by the direction of the arrow and the magnitude of the vector given by the length of the line segment. A vector has an initial point A and a te→minal point B and is often represented symbolically in the form AB. Graphically vectors can be added, subtracted, and multi- plied by a scalar. For our purposes, an analytic formulation of a vector is re- 3 2 quired in R (R ). This formulation is obtained by subtract- ing the coordinates of its initial point, A = (0 ;y0;z0) from the coordinates of its terminal→point, B = (1 ;1 ;1 ), with the resulting vector represented as AB= ⟨x 1 x ;0 −1y ;z 0 z1⟩: 0 → The magnitude or norm or length of the vector AB, repre- → sented symbolically as | AB |, is given by the distance between its initial and terminal point using the formula → | AB | = Example: Find the representation and length of a vector with initial point (1;−2;3) and terminal point (4;0;−3). Example ▯nd the length of the vector v = ⟨3;−7;1⟩: Vectors expressed with respect to a coordinate system may be added, subtracted, and multiplied by a scalar using the following rules: v = ⟨x ;1 ;1 ⟩1 w = ⟨x ;y 2z ⟩2 c2∈ R i. v + w = ii. v − w = iii. cv = Example: If v = ⟨3;−7;1⟩ and w = ⟨−2;0;3⟩, ▯nd 2v − 3w. Properties of Vector Operations The following properties hold for vectors u, v, and w and scalars a and c. 1. u + v = 2. (u + v) + w = 3. v + 0 = 4. v + (−v) = 5. c(u + v) = 6. (a + c)v = 7. 0v = 8. c0 = 9. 1v = 10. a(cv) = A unit vector is a vector of length one. If v is not the zero vector, then a unit vector in the direction of v is given by ^ = Example: Find the unit vector in the direction of the vector w = ⟨−2;0;3⟩. The standard unit vectors are given by: i = e1= ⟨1;0;0⟩, j = ^e2= ⟨0;1;0⟩, and k = ^ e3= ⟨0;0;1⟩. These vectors point in the direction of the positive x, y, and z coordinate axes respectively. Any vector in R can be represented as a sum of scalar mul- tiples of the standard unit vectors: ⟨a;b;c⟩ =.
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