Begin2.DVI

6-9. Consider the triangle defined by the three vertices (6 , 0 ,0),(0 , 6 ,0) and (0 , 0 ,12).

Use vector methods to find the area of the triangle.

6-10. Let the sides of a quadrilateral be

denoted by the vectors A,
B ,
C ,
D
such that

A
+B
 +C
+D
 =
 0.

Use vectors to show that the lines joining the

midpoints of the sides of this quadrilateral form

a parallelogram.

6-11. Let
r 0 represent the position vector of the center of a sphere of radius ρand

let
r represent the position vector of a variable point on the surface of the sphere.

Find the equation of the sphere in a vector form. Simplify your result to a scalar

form.

6-12. For A
=ˆe 1 + 2 ˆe 2 + 2 ˆe 3 and B
= 7 ˆe 1 + 4 ˆe 2 + 4 ˆe 3

(a)Find a unit vector in the direction of B
.

(b)Find a unit vector in the direction of A
.

(c)Find the projection of A
on B
.

(d)Find the projection of B
on A
.

6-13. (a) Find a unit vector perpendicular to the vectors

A
=ˆe 1 −ˆe 2 +ˆe 3 and B
=ˆe 1 +ˆe 2 −ˆe 3

(b) Find the projection of B
on A
.

6-14. For A
=−ˆe 1 +

√

3 ˆe 2 +

√

5 ˆe 3 and ˆeα= cos αˆe 1 + sin αˆe 2

(a) Verify that ˆeαis a unit vector for all α.

(b) Find the projection of A
on ˆeα.

(c) For what angle αis the projection equal to zero?

(d) For what angle αis the projection a maximum?

6-15. Assume A
(t) has derivatives of all orders. Find the constant vectors

A
0 ,A
1 ,... , A
n,... if

A
(t) = A
 0 +A
 1 (t−t^0 )

1! +

A
 2 (t−t^0 )

2

2! +···+

A
n(t−t^0 )

n

n! +···

Hint: Evaluate A
(t)at t=t 0 , then differentiate A
(t)and evaluate result at t=t 0.