Begin2.DVI
Example 6-4. Given the vectors A= 2 ˆe 1 + 3 ˆe 2 + 6 ˆe 3 and B =ˆe 1 + 2 ˆe 2 + 2 ˆe 3 Find: (a) |A|, |B|, A·B, |A+B| ...
Example 6-5. (The Schwarz inequality) Show that for any two vectors A and B one can write the Schwarz inequality |A·B |≤| A ...
or |A+B|^2 =|A|^2 +2(A·B)+ |B |^2 ≤| A|^2 +2 |A·B |+|B|^2 (6 .19) Using the Schwarz inequality |A·B |≤| A|| B |the ...
Properties of the Cross Product A×B =−B×A (noncommutative) A×(B+C) = A×B+A×C (distributive law) m(A×B) = (mA)×B = ...
Geometric Interpretation A geometric interpretation that can be assigned to the magnitude of the cross product of two vectors is ...
A physical interpretation can be assigned to the triple scalar product A·(B×C)is that its absolute value represents the volum ...
Solution Use the determinant form for the cross product and express the triple scalar product as a determinant as follows. A·(B ...
In a similar fashion one can show that the vectors (A×B)×C, Aand B are coplanar so that there exists constants γand δsuch t ...
Example 6-10. Derive the law of sines for the triangle illustrated in the figure 6-10. Solution The sides of the given triangle ...
Example 6-11. Derive the law of cosines for the triangle illustrated. Figure 6-11. Triangle for law of cosines. Solution Let C ...
where λis a scalar parameter. Note that as λvaries from 0 to 1 the position vector r moves from r 1 to r 2 .An alternative fo ...
Define the unit vector ˆeα=|α^1 |αand construct the line from (x 0 , y 0 , z 0 )which is perpendicular to the given line and l ...
Construct the vector r 0 −r 1 which points from the terminus of r 1 to the terminus of r 0 and construct the unit normal to ...
The total moment about the x-axis is therefore the sum of these moments and given by M 1 =F 3 y 1 −F 2 z 1. (b) For the moment a ...
The total moment about the origin is a vector quantity represented as the vector sum of the above moments in the form M 0 =M 1 ...
moment about the line L is then the projection of the vector moment MB on this line. If ˆeLis a unit vector along the line, the ...
where t represents some convenient parameter, say time. The derivative of the position vector r with respect to the parameter t ...
Example 6-18. Consider the space curve defined by the po- sition vector r =r (t) = cos teˆ 1 + sin tˆe 2 +tˆe 3. This curve sw ...
Qand the vector ∆r representing the direction of the secant line through the points P and Q. Letting the point Qapproach the po ...
where the components ui(t),vi(t)and wi(t), i = 1, 2 , 3 are continuous and differentiable, the following differentiation rules c ...
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