Vectors Flashcards
Vector cross product
^ a x b = |a| |b| sinθ n θ is the angle between vectors a,b |a| |b| sinθ is the magnitude of the vector Finds a vector perpendicular to a and b
Does the order matter cross product
Yes, a x b = -(b x a)
Right-hand rule
Thumb, index, middle fingers at right angles
Thumb = a x b
Index = a
Middle = b
Conditions for a x b = 0
a,b are parallel or one is 0
i x j
k
k x j
-i
i x k
-j
Cross product method for vectors a(a1, a2, a3) and b(b1, b2, b3)
i j k |
| a1 a2 a3 |
| b1 b2 b3 |
Area of a triangle from 2 directions
1/2 |a x b|
Can find a,b if you have 3 points
Area of a parallelogram from the 2 directions
|a x b|
a, b are not parallel sides
Can find a,b with vectors between vertices
a • (b x c)
a(a1, a2, a3), b(b1, b2, b3), c(c1, c2, c3)
a1 a2 a3 |
| b1 b2 b3 |
| c1 c2 c3 |
Triple scalar product facts
Reordering a, b, c has no effect as long as a,b,c cycle is maintained, else result is negated
a. (a x b) = 0
Volume of a parallelepiped
|a • (b x c)|
h = |a| cosθ (θ is angle between vertical and edge with vertical component)
area of base = |b x c|
V = |b x c| |a| cosθ
Parallelepiped
A prism with parallelograms for all 6 faces
Equation of a line with cross product
r x b = a x b (r is a general point on the line, a is a known point on the line, b is direction)
Line equation derivation
r - a = b (from known point)
r = a + λb (from general point)
b is parallel to λb so (r- a) x b = 0
expand tp r x b = a x b
Cosines
cos^2 α + cos^2 β + cos^2 γ = 1
α, β, γ are the angles between the vector a and the i, j, k axes respectively
Equation of a plane cross product method
Identify two vectors parallel to the plane
n is the cross product of these
r . n = a . n
Intersection of planes
Use cross product to find a vector perpendicular to both planes (direction, d)
Let x = 0 and solve the plane equations simultaneously for a point (a)
Use r = a + λd
Shortest distance between skew lines r = a + λb and r = c + μd
(a - c) • (b x d) |
| ——————– |
| | b x d | |
Shortest distance between parallel linesr = a + λb and r = c + μb
b x (a - c) |
| ————–|
| |b| |
Volume of a tetrahedron
1/6 |a • (b x c)|
