Vectors

Question 1

Properties of triple scalar product

Answer 1

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

Question 2

Volume of parallelepiped and tetrahedron

Answer 2

parallelepiped = base x ppd height
= a.(bxc) for non parallel vectors

tetrahedron = 1/3 base x height
1/6 a.(bxc)

Question 3

Deriving projection of a vector a onto b (called u)
angle θ between them

Answer 3

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|

Question 4

Determining direction of axb

Answer 4

right hand
first finger a, second finger b, axb in direction of thumb

Question 5

easy way to remember unit cross products

Answer 5

Start w ixj=k imagine moving along, ie kxi=j and kxj=i

swap order and u get negative

Question 6

Properties of vector product

Answer 6

Distributive – ax(b+c) = (axb) + (bxc)

Question 7

When best to use diff averages

Answer 7

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

Question 8

Rules for finding quartiles for discrete data

Answer 8

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

Question 9

Rules for finding quartiles for grouped data
+ assumptions in interpolation

Answer 9

Q1 is n/4th val
Q2 is n/2th
Q3 is 3n/4

Question 10

Measures of spread

Answer 10

Range - Difference between largest and smallest values in dataset

IQR - Difference between upper and lower quartile

Interpercentile range - Difference between 2 given percentiles

Question 11

Direction cosine notes

Answer 11

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

Question 12

Proof that l₁l₂+m₁m₂+n₁n₂ = cosθ for any two intersecting lines

Answer 12

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)