WTW164 Lu1.1 - Lines and PLanes

Give the most general format of a vector

A bar = <a, b, c>

How is a vector formed from 2 points in 3D space

Subtract the end point x,y and z values from the initial points x,y and z values

Eg.) Take A = (x,yz) and b = (x2, y2, z2)

Then V bar = <x2-x, y2-y, z2-z>

Can you describe how to determine a vector using a vector equation.

You need to take two vectors a and d. And lets vector A be the position vector and vectro d be the direction vector. The final vector equation would then be

V = a bar + t x d bar

Can you expand on the most general forms of a vector equation into its other forms

V bar = <x, y, z> = a bar + t x d bar

Therefor <a,b,c> + t<u,v,w> is an expanded form

If we expand further x = a +ut, y = b + vt and z = c +wt (parametric equations)

Making t the subject of the equation we get

T = x-a/u etc and setting all ts equal we finally get

X-a/u = y-b/v = z-c/w (symetric equation)

Since we are now in 3 dimension describe all the behaviours of lines in 3D

1.) Lines intersect where there exists a point where their two parametres equate

2.) Lines are parallel if their direction vectors are scalar multiples

3.) They are skew if the above two are disproved