Vectors Flashcards
Properties of triple scalar product
it’s cyclic, ie
a.(bxc) = b.(cxa) = c.(axb)
if a vector is repeated then is equal to 0
a.(axp) or p.(axa) for any p = 0
Volume of parallelepiped and tetrahedron
parallelepiped = base x ppd height
= a.(bxc) for non parallel vectors
tetrahedron = 1/3 base x height
1/6 a.(bxc)
Deriving projection of a vector a onto b (called u)
angle θ between them
u = û |u|
Consider geometry - draw right angle triangle from a onto b, st a is hypotenuse — |u| =|a|cosθ
additionally, cosθ= (a.b)/|ab|
eliminating cosθ – |u| = (a.b)/|b|
u is parallel to b, share the same unit vector
b(unit) = b/|b| = unit vector of u
altogether u = (a.b) x b/|b^2|
Determining direction of axb
right hand
first finger a, second finger b, axb in direction of thumb
easy way to remember unit cross products
Start w ixj=k imagine moving along, ie kxi=j and kxj=i
swap order and u get negative
Properties of vector product
Distributive – ax(b+c) = (axb) + (bxc)
When best to use diff averages
Mode - When data is qualitative, or when is quantitative with one or 2 modes. Not useful if each value occurs once
Mean - Used for quantitative data, therefore giving a true measure, however affected by extremes
Median - for quantitative, usually when there are extremes as they don’t affect median
Rules for finding quartiles for discrete data
Q1 = n/4, if it’s a whole number, then Q1 is halfway between this data point and the next. If not, round up and pick this data point
Q3 = 3n/4 same principles
Rules for finding quartiles for grouped data
+ assumptions in interpolation
Q1 is n/4th val
Q2 is n/2th
Q3 is 3n/4
Measures of spread
Range - Difference between largest and smallest values in dataset
IQR - Difference between upper and lower quartile
Interpercentile range - Difference between 2 given percentiles
Direction cosine notes
Line parallel to vector a=xi+yj+zk
has direction ratios x:y:z and direction cosines x/|a|, y/|a| and z/|a| written as l,m and n
sum of squares of direction cosines =1
line can be written as x-x1/l = y-y1/m = z-z1/n
direction cosines are in same ratio as ratios
Proof that l₁l₂+m₁m₂+n₁n₂ = cosθ for any two intersecting lines
Given any pair of intersecting lines with direction vectors s1 and s2 divide each direction vector
by a constant to produce direction vectors r1 = x1+y1+z1 and r2= x2+y2+z2 st their magnitudes are 1
l1 = x1/|r1| = x1 —- m1=y1 —- n1= z1
similarly for l2, then use dot product formula to equate expressions.
(can also be done without dividing just keep in terms of modulus)
Condition for 3 vector to be coplanar
(axb).c = 0
