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By Jazar R.N.

ISBN-10: 0470398353

ISBN-13: 9780470398357

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51) Example 14 Linear Dependence and Independence The n vectors r1 , r2 , r3 , . . , rn are linearly dependent if there exist n scalars c1 , c2 , c3 , . . 52) The vectors r1 , r2 , r3 , . . , rn are linearly independent if they are not linearly dependent, and it means the n scalars c1 , c2 , c3 , . . 56) and if c2 = 0, we have which shows r1 and r2 are colinear. 57) where at least one of the scalars c1 , c2 , c3 , say c3 , is not zero; then r3 = − c1 c2 r1 − r2 c3 c3 which shows r3 is in the same plane as r1 and r2 .

We may show a −→ − → vector by an ordered pair of points with an arrow, such as OP . The sign PP indicates a zero vector at point P. Length and direction are necessary to have a vector; however, a vector may have five characteristics: 1. Length. The length of section OP corresponds to the magnitude of the physical quantity that the vector is representing. 2. Axis. A straight line that indicates the line on which the vector is. The vector axis is also called the line of action. 3. End point. A start or an end point indicates the point at which the vector is applied.

65) This is the length of the projection of r on l . 9 A line l and a vector r intersecting at the origin of a coordinate frame. 2 Vector Algebra 19 So, the inner product can be used to find the projection of a vector on a given line. 74) There is no meaning for a vector with a negative or positive odd exponent. 81) then either a = 0 or b = 0, or a and b are perpendicular. 86) An alternative method is decomposition of the vector equation along the axes ıˆ, ˆ, kˆ of the coordinate frame and solving a set of three scalar equations to find the components of the unknown vector.

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Advanced Dynamics by Jazar R.N.

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