Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
Problems
1.1. Given the vector A ˆ…2; 2; 1† and B ˆ…6; 3; 2†, determine:
(a) 6A 3B, (b) A 2 ‡ B 2 ,(c) A B, (d) the angle between A and B, (e) the
direction cosines of A, (f ) the component of B in the direction of A.
1.2. Find a unit vector perpendicular to the plane of A ˆ…2; 6; 3† and
B ˆ…4; 3; 1†.
1.3. Prove that:
(a) the median to the base of an isosceles triangle is perpendicular to the
base; (b) an angle inscribed in a semicircle is a right angle.
1.4. Given two vectors A ˆ…2; 1; 1†, B ˆ…1; 1; 2† ®nd: (a) A B, and (b) a
unit vector perpendicular to the plane containing vectors A and B.
1.5. Prove: (a) the law of sines for plane triangles, and (b) Eq. (1.16a).
1.6. Evaluate …2^e 1 3^e 2 †‰…^e 1 ‡ ^e 2 ^e 3 †…3^e 1 ^e 3 †Š.
1.7. (a) Prove that a necessary and sucient condition for the vectors A, B and
C to be coplanar is that A …B C† ˆ0:
(b) Find an equation for the plane determined by the three points
P 1 …2; 1; 1†, P 2 …3; 2; 1† and P 3 …1; 3; 2†.
1.8. (a) Find the transformation matrix for a rotation of new coordinate system
through an angle about the x 3 …ˆ z†-axis.
(b) Express the vector A ˆ 3^e 1 ‡ 2^e 2 ‡ ^e 3 in terms of the triad ^e 1^e 0 2^e 0 3 0 where
the x1x 0 2 0 axes are rotated 458 about the x 3 -axis (the x 3 -andx3-axes
0
coinciding).
1.9. Consider the linear transformation Ai 0 ˆ P3
jˆ1 ^e i 0 ^e j A j ˆ P3
jˆ1 ijA j . Show,
using the fact that the magnitude of the vector is the same in both systems,
that
X 3
iˆ1
ij ik ˆ jk … j; k ˆ 1; 2; 3†:
1.10. A curve C is de®ned by the parametric equation
r…u† ˆx 1 …u†^e 1 ‡ x 2 …u†^e 2 ‡ x 3 …u†^e 3 ;
where u is the arc length of C measured from a ®xed point on C, andr is the
position vector of any point on C; show that:
(a) dr=du is a unit vector tangent to C;
(b) the radius of curvature of the curve C is given by
2
ˆ 4
!
d 2 2 ! 2 ! 3 2
x 1
du 2 ‡ d2 x 2
du 2 ‡ d2 x 3 5
du 2
1=2
:
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