Exam 1

Question 1

1.1

i. Compute the distance from the point P = (2, 1, −5) to the closest point on the sphere defined by the equation
x^2 + y^2 + z^2 − 6y + 9 = 0.
ii. Let the situation be as above, and let Q be the furthest point on the sphere from P. Compute vector PQ

Answer 1

ii. use unit vector to add the radius to get direction too

Question 2

1.2

Find the center and radius of the sphere defined by the equation
x^2 − 6x + y^2 + 10y + z^2 + 2z = 1

Question 3

1.3

Consider the sphere given by the following equation:
___________
Find the value of a so that the sphere has radius 5.

Question 4

1.3

Consider the sphere given by the following equation:
_____
The sphere has radius 5. Find the y-coordinate of the center.

Question 5

what method do you use for orthogonal vectors

Answer 5

dot product=0

Question 5

2.2

A 500lb load hangs from three cables of equal length that are
anchored at the points

Find the vector describing the force on the cable anchored at B, i.e.,
the force the cable is exerting to pull the point B (resp. C, D) in the
direction of the point A.

Answer 5

1. *load is at A. find vectors BA, CA, and DA
2. find magnitude of any of them; they are all the same so just pick one
3. T1(1/magnitude)<>
4. T2(1/magnitude)<>
5. T3(1/magnitude)<>
6. • all of them together = <0,0,load weight>
7. T1(1/magnitude)<last>+T2(1/magnitude)<last>+T3(1/magnitude)<last></last></last></last>
8. solve for T1=T2=T3=x
9. answer: x(1/magnitude)<>

Question 6

what do you do if you know it’s a unit vector

Answer 6

set the magnitude = 0

Question 6

3.2. Determine the angle between the following two tangent lines:
one is to the curve y = 3x at point (1, 3) and the other is to the curve
y = 3x
5 at point (1, 3)

Answer 6

dot product formula
<1, slope at point>

Question 7

Determine the constant c so that the following three vectors are
coplanar:

~a = h7, c, 1i
~b = h0, 3, −1i
~c = h−3, 4, −2i.

Answer 7

coplanar if AB dot (AC x AD)=0

Question 8

Find the area of the triangle formed by the following three points
P = (1, 0, 1),
Q = (−2, 1, 3),
R = (4, 2, 5).

Answer 8

find magnitude of PQ and PR and then plug into 1/2bh, use cross product
* cross product then magnitude
* not magnitudes only

Question 8

what do you use to find area

Answer 8

cross product

Question 9

Write inequalities which describe the geometric objects below.
(i) The solid cylinder whose central axis is the line given by the
equations x = 3, y = −5, and the cross section perpendicular
to the axis is a disk of radius 2.
(ii) The solid upper hemisphere of the sphere of radius 7 centered
at (1, −2, 3). (We choose the z-axis to be the verical one, while
the xy-plane is horizontal. The word “upper” is with respect
to the vertical z-axis.)

Answer 9

use equation of a circle for i and equation of a sphere for ii