math guide - 34.41

34.3.4Planes, Lines, etc.

•The most fundamental mathematical geometry is a line. The basic relationships are given below,

y =	mx + b		defined with a slope and intercept
mperpendicular =		1	a slope perpendicular to a line
		---
		m
m =	y2 – y1		the slope using two points
	---------------
	x2 – x1

x	y	= 1	as defined by two intercepts
--	+ --
a	b

• If we assume a line is between two points in space, and that at one end we have a local reference frame, there are some basic relationships that can be derived.

math guide - 34.42

y

θ β

d

( x2, y2, z2)

( x1,

y1, z1)

d = ( x

2 – x1)

2

+ ( y2 – y1)

2

+ ( z2 – z1)

2

θ γ

θ α

x

z

( x0, y0, z0)

The direction cosines of the angles are,

x2 – x1

y2 – y1

z2 – z1

---------------

θ β

=

---------------

--------------

θ α = acos

d

acos

d

θ γ = acos

d

( cos θ

α

) 2 + ( cos θ

) 2

+ ( cos θ

) 2

= 1

β

γ

The equation of the line is,

x – x1

y – y1

=

z – z1

Explicit

---------------

= ---------------

--------------

x2 – x1

y2 – y1

z2 – z1

( x, y, z) = ( x1, y1,

z1) + t( ( x2, y2, z2) – ( x1, y1, z1) )

Parametric t=[0,1]

• The relationships for a plane are,

math guide - 34.43

The explicit equation for a plane is,

Ax + By + Cz + D = 0

where the coefficients defined by the intercepts are,

A =	1	B =	1	C =	1	D = –1
	--		--		--
	a		b		c

y
b
	P1 = ( x1, y1, z1)
	N
	P2 = ( x2, y2, z2)
P3	= ( x3, y3, z3)
	a	x

z

c

The determinant can also be used,

x – x1 y – y1 z – z1

det

x – x

2

y – y

2

z – z

2

= 0

x – x3 y – y3 z – z3

det

y2 – y1 z2 – z1

( x – x ) + det

z2 – z1 x2 – x1

( y – y )

y3 – y1 z3 – z1

1

z3 – z1 x3 – x1

1

+ det

x2 – x1 y2 – y1

( z – z ) = 0

x3 – x1 y3 – y1

1

The normal to the plane (through the origin) is,

( x, y, z) = t( A, B, C)

34.3.5Practice Problems

1.What is the circumferenece of a circle? What is the area? What is the ratio of the area to the circumference?

math guide - 34.44

2.What is the equation of a line that passes through the points below?

a)(0, 0) to (2, 2)

b)(1, 0) to (0, 1)

c)(3, 4) to (2. 9)

3.Find a line that is perpendicular to the line through the points (2, 1) and (1, 2). The perpendicular line passes through (3, 5).

4.Manipulate the following equations to solve for ‘x’.

a)x2 + 3x = –2

b)sin x = cos x

(ans.
a)	x2 + 3x		= –2
	x2 + 3x + 2 = 0
	x =	– 3 ± 32 – 4( 1) ( 2)		=	– 3 ± 9 – 8	=	– 3 ± 1	= –1, –2
		------------------------------------------------	2( 1)		-----------------------------		----------------
					2		2
b)	sin x	=	cos x
	----------sin x	=	1
	cos x
	tan x	= 1
	x =	atan 1

x= … , –135° , 45° , 225° ,…

5.Simplify the following expressions.

a)		( cos 2θ ) 2
sin 2θ		----------------------	+ sin 2θ
		sin 2θ

(ans.

a)		( cos 2θ ) 2		2	2
	sin 2θ	----------------------	+ sin 2θ	= ( cos 2θ ) + ( sin 2θ )		= 1
		sin 2θ