chapter 11

Question 1

dot product of u and v

Answer 1

\u\v\cos(theta)

magnitudes times the cos of angle inbetween

Question 2

how to determine if vectors are orthogonal

Answer 2

their dot product will be zero

Question 3

algebraec dot product

Answer 3

u dot v = u1v1 + u2v2 +u3v3

Question 4

orthogonal projection of u onto v

Answer 4

udotv/magv

Question 5

Work

Answer 5

f DOT D

Question 6

ijk circle

Answer 6

ijk arrows pointing left to right

Question 7

cross product

Answer 7

\u\v\sin(theta)

Question 8

testing paralell

Answer 8

uxv must be 0

0 cross product

Question 9

evaluating a cross product

Answer 9

use determenants and shit

Question 10

torque

Answer 10

r x F

remember that the angle must be the angle invetween

Question 11

2 things i need for a line in 3d space

Answer 11

point and a vector multiplied by time

Question 12

tangent vector at a point t

Answer 12

derive y y and z then.

plug t into the original to find the point

Question 13

when are vectors teh same

Answer 13

same direction and magnitude

Question 14

when are 2 vectors paralell

Answer 14

their vectors are scalar multiples of eachother

Question 15

find a vector perpendicular to another vector

Answer 15

let the known vector be P=ai+bj+ck…………………….(1)
and, let the unknown vector be Q=xi+yj+zk………………(2)
Since the two vectors are to be perpendicular to each other,their dot product should be 0.
ie : P.Q=0=(ai+bj+ck).(xi+yj+zk)=ax+by+cz=0………(3)
Now we have three variables and one equation. So there exists infinitely many solutions.
To find one of them, assign any value to any two variables of x,y and z.
This will give you the third variable when you solve the above equation.
Then you get a vector when you plugin the values of x,y and z to the Q
equation (2).
then you have found a vector which satisfies the condition given in the question.
You may find vectors of any magnitude that still satisfies the condition by multiplying a suitable scalar to the newly found vector Q.

Question 16

difference between a sphere and a ball

Answer 16

rather than an = sign there will be less than radius

Question 17

Area of a triangle or paralellogram

Answer 17

.5uvsintheta triangle

uvsintheta paralellogram

Question 18

using dot products how do i get a 90 degree angle

Answer 18

u dot v = 0

Question 19

using dot products how do i get an obtuse degree angle

Answer 19

u dot v is less than 0

Question 20

using dot products how do i get an acute degree angle

Answer 20

u dot v is greater than 0

Question 21

what is the arc length for a vector based parametric function

Answer 21

given
int atob sqrt(f’^2+g’^2+h’^2)
also just the int atob
of the magnitude of derivative of the r vector.

Question 22

arc length of a polar curve

Answer 22

int a to b sqrt(f^2+f’^2)