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 equation of the line.
6-25.
(a) Derive the vector equations of a line in the following forms.
(r −r 1 )×(r 2 −r 1 ) = 0 and r =r 1 +λ(r 2 −r 1 )
for a line passing through the two points P 1 (x 1 , y 1 , z 1 )and P 2 (x 2 , y 2 , z 2 ).
(b) Show these vector equations produce the same scalar equations for deter-
mining points on the line.
6-26. Sketch the vectors ˆeα= cos αˆe 1 +sin αˆe 2 and ˆeβ= cos βˆe 1 +sin βeˆ 2 assuming
αand β are acute constant angles.
(a) Show ˆeαand ˆeβare unit vectors.
(b) From the dot product ˆeα·ˆeβ derive the addition formula for cos(β±α)
(c) From the cross product ˆeα׈eβ,derive the addition formula for sin(β±α)
6-27. Verify that ˆei׈ej=±ˆekwhere the +sign is used if (ijk )is an even permu-
tation of (123) and the −sign is used if (ijk )is an odd permutation of (123).
(a) Verify the above by taking three consecutive numbers from the set
{ 1 , 2 , 3 , 1 , 2 , 3 } for the values of i, j, k. These are called the even permutations
of the numbers (123).
(b) Verify the above by taking three consecutive numbers from the set
{ 3 , 2 , 1 , 3 , 2 , 1 }for the values of i, j, k. These are called the odd permutations of
the numbers (123).
6-28. Let α 1 , β 1 , γ 1 and α 2 , β 2 , γ 2 be the direction angles of two lines. Move each
line parallel to itself until it passes through the origin. The angle between two lines
is defined as the angle between the shifted lines, which pass through the origin.
(a) Show that the angle θbetween two lines can be expressed in terms of the direction
cosine of the lines and
cos θ= cos α 1 cos α 2 + cos β 1 cos β 2 + cos γ 1 cos γ 2.(b) Find the angle between the lines defined by the equations
r = (1 + 2t)ˆe 1 + (1 + t)ˆe 2 + (1 + 2t)ˆe 3 and
r = (1 + 2t)ˆe 1 + (2 + 2 t)ˆe 2 + (6 + t)ˆe 3.