vectors Flashcards
how to test if two vectors are orthogonal
the dot product will be 0 as the angle is 90 degrees
scalar product
a . b = |a||b|cosx
unit vector
u/|u|
vector product
determinant of a matrix
what angle will be between two parallel vectors
0 or pie
sinx formula
|a x b|= |a||b|sinx
what does doing the vector product find
the area
finds the normal of two vectors
triple scalar product
(a x b) . c
when are 3 vectors coplanar
(a x b) . c = 1
triple vector product
a x (b x c)
Finds volume
equation of a line
r = a + tu
how do you find u for the equation of a line
b - a
distance formula
y = sqrt((x1 - x2) + (y1 - y2))
vector equation of a plane
( r - a) . n = 0
how to find line POI
- write the 2 line equations in cartesian form
- solve for t
- substitute t into cartesian equation
- this gives the POI
how to find plane POI
- find cartesian equation of the line
- sub these values in the equation of the plane
- find the value of t
- if t disappears the line and plane are parallel
- sub t into equation of the line to get POI
angle between a line and a plane
use the dot product of direction vector of the line and normal of the plane
angle between 2 planes
cos x = n1 . n2 / |n1||n2|
- if n1 . n2 is positive the angle is acute so the angle is pie/2 - angle
- if n1 . n2 is negative the angle is obtuse so the angle is angle - pie/2
how to find the minimum distance with a point and a plane
- write line equation in cartesian form
- sub these into plane equation
- find t and sub into cartesian line equation
- this is the POI
- use the original point and the POI and find the distance between them
how to find the minimum distance with 2 planes
- find a point on one P1 by substituting y=z=0
- find the line equation normal to P2 passing through the point we found
- do the same process as line and plane
