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
(b) B×A
(c) A·B
(d) (A+B)×A
(e) The angle between A and B
(f) 3 A× 2 B
(g) (A+ 3B)×B
(h) (B−A)×(B+A)
(i) A·(A+B)
6-17. The sides of a parallelogram are A=ˆe 1 +2 ˆe 2 +2 ˆe 3 and B= 2 ˆe 1 +9 ˆe 2 +2 ˆe 3.
(a) Find the vectors which represent the diagonals of this parallelogram.
(b) Find the area of the parallelogram.
6-18. Determine the direction cosines of the vector ρ=
√
2 eˆ 1 +ˆe 2 −ˆe 3.
6-19. Explain why two vectors are said to be linearly dependent if their vector
cross product is the zero vector.
6-20. Three noncolinear points P 1 (x 1 , y 1 , z 1 ), P 2 (x 2 , y 2 , z 2 ),and P 3 (x 3 , y 3 , z 3 )determine
a plane. Let r 1 , r 2 , r 3 denote the position vectors from the origin to each of these
points, respectively, and let r denote the position vector of any variable point (x, y, z)
in the plane.
(a) Describe and illustrate the vector r 3 −r 1.
(b) Describe and illustrate the vector r 2 −r 1.
(c) Describe and illustrate the vector (r 2 −r 1 )×(r 3 −r 1 ).
(d) Explain the geometrical significance (r −r 1 )·[(r 2 −r 1 )×(r 3 −r 1 )] = 0.
6-21. Find the parametric equations of the given line. Also find the tangent vector
to the given line r = 3 ˆe 1 + 4 ˆe 2 + 2 ˆe 3 +λ(ˆe 1 −ˆe 2 ).
6-22. (a) Find the area of the triangle having vertices at the points
P 1 (0, 0 ,0) P 2 (0 , 3 ,4) P 3 (4, 3 ,0).
(b) Find a unit normal vector to the plane passing through the above three points.
(c) Find the equation of the plane in part (b).
6-23. Distance between two skew lines Let line 1 pass through points P 0 (x 0 , y 0 , z 0 )
and P 1 (x 1 , y 1 , z 1 ). Let line 2 pass through the points P 2 (x 2 , y 2 , z 2 )and P 3 (x 3 , y 3 , z 3 ).
(a) Show N =P 0 P 1 ×P 2 P 3 is perpendicular to both lines. (b) Show the projection of
P 2 P 1 onto N gives the distance between the lines.
