vectors

Question 1

eqns of a plane

Answer 1

scalar product form: r.n =p
vector form: r = a + λb + μc
cartesian form: ax + by + cz = d

Question 2

angle between line and plane

Answer 2

angle between n and d, then 90-

Question 3

angle between two planes

Answer 3

angle between the normal vectors

Question 4

eqns of line

Answer 4

r = a + μd
(r-a) x d = 0
x-x’/a = y-y’/b =z-z’/c

Question 5

cross product

Answer 5

a x b = |a||b|sinx n

n is the unit vector n

Question 6

area of a triangle between two vectors

Answer 6

1/2 |a x b|

Question 7

area of parallelogram between two vectors

Answer 7

|a x b|

Question 8

intersection of a line and plane

Answer 8

plane in cartesian, sub in terms of λ then solve for the point with given λ

Question 9

intersection of two planes

Answer 9

parallel planes = nun and same n
line of intersection, has direction perpendicular to both n, fix some point on the intersection

Question 10

shortest distance between two skew lines

Answer 10

for two lines : r = a + λb and r = c + μd

(a - c).(b x d)/|b x d|
this imagines each line lies on parallel planes and is some multiple (a -c) of the unit vector n

Question 11

shortest distance between parallel lines

Answer 11

fix point on one of lines
general vector from point to other line
direction vector of line . new vector = 0
find || of vector

Question 12

shortest distance between planes

Answer 12

intersect = 0
r . n is the shortest distance to the origin given that n is the unit vector
difference between two distances