VECTORS Flashcards
Define scalar product
a.b = IaIIbI cos ø
Define vector product
a x b = IaIIbI sinø ñ
What does i x i = ?
j x j = k x k = 0
What does
a) i x j = ?
b) k x j = ?
a) k
b) -i
Relationship between vector product and parallel vectors
If a x b = 0, the two vectors are parallel
Vector product, area of triangle
1/2 I a x b I
Vector product area of parallelogram
I a x b I
What does a. b x c mean?
Scalar triple product,
a. (b x c)
Importance of order in scalar triple product
Same cyclic order as vector product
a. b x c = b. c x a
What does a x (b + c) mean? (Distributive law)
a x b + a x c
Volume of a pyramid
1/3 I a. b x c I
Volume of a tetrahedron
1/6 I a. b x c I
Volume of a triangular prism
1/2 I a. b x c I
What does coplanarity mean?
How can it be proven?
If vectors are co-planar, they lie on the same plane
I a. b x c I = 0
Parametric form of equation of a line
r = a + µb
Cartesian/Direction ratio form of equation of a line
x - a1/b1 = y - a2/b2 = z - a3/b3 (=µ)
a from parametric equation = specific point
b = direction
Change to cartesian form of equation of line when direction vector contains a 0
No denomiator, comma instead of = sign
Direction cosines and the angle a parallel line makes with x, y, z axes
using line x - a1/b1 = y - a2/b2 = z - a3/b3
vector b is parallel
angles it makes with the three axes : cos ø =
bi/IbI = direction cosines, denoted as l, m and n
Sum of square of direction vectors = ?
l2 + m2 + n2 = 1
Vector equation form of equation of line
( r - a ) x b = 0
(Because they are parallel)
Equation of plane using two non-parallel vectors on plane
r = a + ßb + µc
b and c are the two non-parallel vectors, a is a specific point on the plane.
Scalar product equation of a plane
r . n = d
d = a . n
a is a point on the plane
n is a vector perpendicular to plane, found by x-ing two direction vectors
Cartesian equation of plane
n1x + n2y + n3z = d
Angle between a line and a plane
90. - the acute angle (180 - angle if not acute) between the line and the normal to the plane calculated using:
cosø = I b .n / IbI InI I
b is direction vector of line
Angle between two planes
Scalar product using the two normals:
I n1 . n2 / In1I In2I I
Shortest distance from point p to line AB?
The perpendicular, using vector product:
I AP x AB I/ IABI
A and B are point on the line
Shortest distance between two lines
- Find any vector AB between two points, one on each line
- x them to find perpendicular vector
- calculate I AB. n I / I n I
Shortest distance point to plane
- Find equation of line that goes through point A perpendicular to the plane
- Find the point of intersection between point on the line and the plane
- Find the distance AP
Intersection of planes x + 2y - z = 2 and 3x -y + 2z = 1
With description
Line of intersection is perpendicular to the normals of both planes
the two normals here are
1 3
2 -1
-1 2
x these to get which is the direction vector
3
- 5
- 7
Substitute values for x, y and z into the the equations of the planes to get a specific point
Intersection between line r = ( 2, -3, 4) + µ(-1, 2, -3) and plane 2x - 3y + z = 6
With description
r = (x, y, z)
x = 2- µ
y = -3 + 2µ
z = 4 -3µ
put these in equation of plane, to get value for µ and then put this into the equation of the plane