Module 1

Question 1

Distance of a Sphere from P to Q

Answer 1

|P-Q| = sqrt [ (a2-a1)^2 + (b2-b1)^2 + (c2-c1)^2 ]

Question 2

SPhere of radius R centered at (a, b, c)

dist^2 = R^2

Answer 2

(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2

Question 3

Definition of a line in 3-D

Answer 3

vector r = < x(t), y(t), z(t) >

where x(t), y(t), and z(t) are linear functions

Question 4

If you find the equation of a line with one point and then do the same process with a different point, you will get:

Answer 4

a different parameterization of the same line

Question 5

r(t) =

Answer 5

= tv = t< a, b, c > = < at, bt, ct >

Question 6

How to find the vector parametric equations for a line?

Answer 6

Use the two points P and Q to find PQ (Q-P). That vector is then the vector parameteriaztion and one of the points P or Q is the point vector.

Question 7

perpendicular to the xz plane:

Answer 7

< 0, 1, 0 >

Question 8

perpendicular to the xy plane:

Answer 8

< 0, 0, 1 >

Question 9

perpendicular to the yz plane:

Answer 9

< 1, 0, 0 >

Question 10

How do you find if two lines intersect?

Answer 10

set one variable of t equal to s. set up two systems of equations and set two variables equal to each other. find t and s. plug t and s into the last variable equal equation.

if the final equations do not match the lines do not intersect.

If they do intersect, plug in t or s into the corresponding r1 or r2 equation to find the point.

Question 11

Dot Product

V . W

Answer 11

= a1a2 + b1b2 + c1c2

Question 12

V . V

Answer 12

||V||^2 = || < a, b, c > || = sqrt(a^2 + b^2 + c^2)

Question 13

cos (pheta)

Answer 13

= ( V . W) / (||V|| ||W||)

to find the angle: multipluy this my the inverse cosine

Question 14

two non-zero vectors are called:

Answer 14

normal/ perpendicular/ orthongonal

Question 15

acute angle =

Answer 15

V . W > 0

cos (pheta) > 0

Question 16

obtuse angle =

Answer 16

V . W < 0

cos (pheta) < 0

Question 17

right angle =

Answer 17

V perpedicular to W -> V . W = 0

Question 18

how to find a vector normal (perpendicular) to a plane:

Answer 18

a(x-x0) + b(y-y0) + c(z-z0) = 0 or ax + by + cz = d

< a, b, c > for both situations

Question 19

Cross Product

Answer 19

< bf - ec, cd - fa, ae - db >

|abc/def| start in second column

Question 20

3 x 3 Determinant

Answer 20

(aei + bfg + cdh) - (gec + hfa + idb)

|abc/def/ghi| start in first column all the way to the third.

Question 21

unit vector in the firection of V : V / ||V|| =

Answer 21

vector e sub vector v

Question 22

parallel to vector V =

Answer 22

“scalar multiple of vector V”

Question 23

Projection of vector U onto vector V: U sub || V =

Answer 23

(U . V / V . V) (V)

V . V = ||V||^2

Question 24

Component/ scalar component of U along V =

Answer 24

U . V / ||V||

The absolute value of this is equal to the length of the projection vector

Question 25

Work (W) =

Answer 25

F . d

units are ft/lbs unless specified otherwise

Question 26

1.

||V x W|| =

Answer 26

||V|| ||W|| sin (pheta)

Question 27

W x V =

Answer 27

• (V x W)

Question 28

V x V =

Answer 28

0

Question 29

V x W = 0 if:

Answer 29

V and W vectors are parallel {||V|| ||W|| sin (pheta) } or one or both of them are vector 0

Question 30

i =

Answer 30

< 1, 0, 0 >

Question 31

j =

Answer 31

< 0, 1, 0 >

Question 32

k =

Answer 32

< 0, 0, 1 >

Question 33

to determine a cross product of i, j, and k, take the shortest parth from 1st to 2nd (i, j, k in clockwise rotation)

Answer 33

if clockwise: positive, if counterclockwise: negative

Question 34

parallelogram spanned by V and W has area:

Answer 34

A = ||V x W||

Question 35

parallelpiped spanned by U, V, and W has volume:

Answer 35

V = | U. (V X W) | = 3X3 determinant

just need absolute value. order does not matter!

Question 36

to find a plane you need a:

Answer 36

point on the plane and a normal vector to the plane

two vectors are perpendicular if their cross product is equal to zero!!

Question 37

vector form

Answer 37

n . < x, y, z> = d = ax0 + by0 + cz0

Question 38

scalar form

Answer 38

a(x-x0) + b(y-y0) + c(z-z0) = 0
ax + by + cz = d (standard form)