Begin2.DVI
6-16. Given the vectors A=eˆ 1 − 2 ˆe 2 + 2 ˆe 3 and B = 3 ˆe 1 + 2 ˆe 2 + 6 ˆe 3 Evaluate the following quantities: (a) A×B ...
6-24. Is the point (6 , 13 ,12) on the line which passes through the points P 1 (1 , 0 ,1) and P 2 (3 , 5 ,2)? Find the equatio ...
6-30. If A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 eˆ 3 and B=B 1 ˆe 1 +B 2 eˆ 2 +B 3 ˆe 3 show that A×B =−B×A. 6-31. If A×B = 0 and ...
6-39. In a rectangular coordinate system a particle moves around a unit circle in the plane z= 0 with a constant angular veloci ...
6-46. For u =u (t) = t^2 ˆe 1 +tˆe 2 + 2tˆe 3 and v =v (t) = t^3 ˆe 1 +t^2 ˆe 2 +t^6 eˆ 3 find the derivatives (a) d dt (u ...
6-54. Evaluate the line integral I= ∫ C F·dr, where F = 3(x+y)ˆe 1 + 5xy ˆe 2 and Cis the curve y=x^2 between the points (0 ...
6-63. For the curves defined by the given parametric equations, find the position vector, velocity vector and acceleration vect ...
6-70. Consider the tetrahedron defined by the vectors A, B, C illustrated. (a) Show the vectors n 1 =^12 A×B, n 2 =^12 B ...
Chapter7 Vector Calculus I One aspect of vector calculus can be described as taking many of the concepts from scalar calculus, g ...
A curve is called an oriented curve if (i) The curve is piecewise smooth. (ii) The position vector r =r (t), when expressed in ...
Example 7-1. Reflection property for the parabola. The parabola y^2 = 4 px with focus F having coordinates (p,0) can be represen ...
in figure 7-1 are the complementary angles to θ 1 and θ 2. These angles are labeled as αand β. Construct the vector r 1 from po ...
Example 7-2. Reflection property of the ellipse. Consider the ellipse x 2 a^2 +y 2 b^2 = 1 having eccentricity e < 1 and foci ...
simultaneously, to obtain t 0 = tan −^1 ( ay 0 bx 0 ) . The derivative vector dr dt =−asin tˆe 1 +bcos tˆe 2 evaluated at the v ...
Using algebra one can verify that the equation (7.7) reduces down to an identity so that the angles αand βare equal. This in tur ...
complementary angles associated with these angles are labeled αand βrespectively. Next construct the vector r 1 running from th ...
for all values of the parameter t. The angles αand βconstructed at the point P can be calculated from the dot products (−ˆet)·ˆe ...
Normal and Binormal to Space Curve Recall that the unit tangent vector to a space curve r =r (t), for any value of the paramet ...
binormal ˆeband unit normal ˆenis called the normal plane. The plane which is per- pendicular to the principal normal ˆenis call ...
which implies that the vector d ˆeb ds is also perpendicular to the vector ˆeb. These two results show that the derivative vecto ...
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