Week 6 - Vector geometry, Functions of many variables

Question 1

Vector equation of a straight line

Answer 1

x = a + tp

where a is the position vector of a point,
p is the displacement vector

Question 2

Angle between 2 vectors

Answer 2

xdoty = ||x|| ||y||cosθ

Question 3

Vector equation of a hyperplane + How to find the Cartesian equation

Answer 3

1. pdot(x - a) = 0
2. pdotx = pdota

Question 4

Distance between a point & a ‘hyperplane’

Answer 4

b - x = λp
pdotx = k

Then apply Pythagoras’ theorem, ||b - x||

Question 5

Directional derivative

Answer 5

Duf(a,b) = unit vector dot gradient vector

Question 6

At the point a, the directional derivative in any direction u is…

Answer 6

Duf(a) = ||gradient vector|| cosθ

Question 7

(αx + βy) dot z = ?

Answer 7

α(x dot z) + β(y dot z)