Exercises
6-1. For the vectors A= 3 ˆe 1 + 2 ˆe 2 +ˆe 3 and B= 6 ˆe 1 −ˆe 2 + 2 ˆe 3 calculate
(a) A+B (b) 6A− 3 B (c) A+ 2 B
6-2. Use vectors to show that the diagonals of a parallelogram bisect one another.
6-3. Use vectors to show that the line segment connecting the midpoints of two
sides of a triangle is parallel to the third side and has one half the magnitude of the
third side.
6-4. In the parallelogram ABCD illus-
trated, construct lines from the vertex Ato the
midpoints of the sides DC and BC. Show that
these lines trisect the diagonal BD.
6-5. Are the given vectors linearly dependent or linearly independent?
(a)A=ˆe 1 +ˆe 2 − 2 ˆe 3
B =− 4 ˆe 1 − 3 ˆe 2
C = 7 ˆe 1 + 6 ˆe 2 − 6 ˆe 3
(b)A= 2 ˆe 1 +ˆe 2 −ˆe 3
B =ˆe 1 −ˆe 2
C = 3 ˆe 3
(c)A= 3 ˆe 1 −eˆ 2 + 2 ˆe 3
B=−ˆe 1 +ˆe 3
C= 14 ˆe 1 − 4 ˆe 2 + 6 ˆe 3.
6-6. If A, B, Care nonzero vectors and A·(B×C) = 0, then determine if the following
statements are true or false.
(i) The vectors A, B, C are linearly independent.
(ii) The vectors A, B, C are linearly dependent.
Justify your answers.
6-7. Let A=A(t)denote a vector which has a constant length Cfor all values of
the parameter t.
(a) Show that A·A=C^2
(b) Show that the derivative vector d
A
dt
is perpendicular to A.
6-8. Show that for r 1 =x 1 ˆe 1 +y 1 ˆe 2 +z 1 ˆe 3 and A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 eˆ 3 the distance
dof an arbitrary point (x 0 , y 0 , z 0 )from the line r =r 1 +tA, is given by
d=|(r 0 −r 1 )׈eA|
where ˆeAis a unit vector in the direction of A and r 0 =x 0 ˆe 1 +y 0 ˆe 2 +z 0 ˆe 3 is a position
vector to the arbitrary point.
