2 - vectors Flashcards
vector eq of a straight line
r = a + λ b
a is the position vector of a point on a line
b is the direction vector parallel to the line
cartesian form of a vector straight line
use (x, y, z) = (a1,a2,a3) + λ (b1, b2, b3)
and simplify to find x y z
x = a1 + λb1
y = a2 + λb2
z = a3 + λb3
rearrange each to have λ as the subject
make λ = λ = λ
if r = a + λb the cartesian form is
x - a1 /b1 = y - a2 /b2 = z - a3 /b3
scalar product/ dot product
a . b = sum of(aibi) = a1b1 + a2b2 + a3b3
the dot product is the sum of the products of the components
angle between 2 vectors
a.b / |a||b| = cos(angle)
vectors have to be pointed away from the angle - if wrong way use other side of the angle
if 2 vectors are perpendicular
a . b = 0
shortest distance from a line to a point
find the distance between the point and the point perpendicular to it on the line
vector / cross product
a x b
five the vector perpendicular to the two other vectors
- put into a 3x3 matrix and find the determinant of each 2x2 when you cover the top row
right hand rule
thumb is the cross product
first finger is a
second finger is b
when you go from a to b it turns left so it screws in so it’s going down
cross product of ijk
ijkijkijkijk…
if you cross along from left to right eg i x j = k the product is positive
if you do it from right to left eg k x j = -i the product is negative
application of cross product
can use to inverse a matrix
(c2 x c3
c3 x c1
c1 x c2)
then divide by det(A)
direction vector
A->B = B - A
hwo to find the vector eq from a cartesian equation
x - a /k = y - b /m = z - c /n
r = (a, b, c) + λ(k, m, n)
intersection of two vector lines
make both equations into one vector with λ and µ in
equate each line
and solve for λ and µ simulaneously
sub back in to find x y z
skew lines
in a plane two lines can not be parallel but also not intersect
- it is not possible to make r1 = r2 and find lambda or mew
