Example: - Find an equation for the plane through Po( - 4, 2, 7) perpendicular to N ?=3i+5j+12k.
Solution:-
N ?=3i+5j+12k
A=3 , B=5 , C=12
D=[3*(-4)+5*2+12*7]=82
3x+5y+12z=82
Example:- Find an equation for the plane through A( 1, 0, -2), B(2,0,3) and C(0,3,0)
Solution:-
(
Home work:-
Example:-find point of intersection the line (x=8+2t , y=-2t ,z=1+t)
With plane (x+2y+3z=6)
Solution
Q(x=8+2t , y=-2t ,z=1+t) substitution line in plane
8+2t+2(-2t)+3(1+t)=6
8+2t-4t+3+3t=6
t+11=6 , t=-5
Q(-2,10,-4)
Angle between planes:-
The angle between planes is defined to be
Angle made by their normal vectors
?=cos^(-1)??((N1.) ?(N2) ?)/(|(N1) ? |.|(N2) ? | )?
Example:- find the angle between the planes
2x-3y-z=10
X+2y-5z=21
Solution (N1) ?=2i-3j-k (N1) ?=i+2j-5k
(N1.) ?(N2) ?=2-6+5=1 |(N1) ? |=?(4+9+1)=?(14 )
|(N2) ? |=?(1+4+25)=?30
?=cos^(-1)??((N1.) ?(N2) ?)/(|(N1) ? |.|(N2) ? | )?=cos^(-1)??1/(?14 ?30)=87.2^0 ?
Example:- find the ve3ctor parallel to the line of intersection of the planes
3x-6y-2z=15 and x+2y-z=5
Solution :-
(N1) ?=3i-6j-2k (N2) ?=i+2j-k
(N1) ?x(N2) ?=|?(i&j&k@3&-6&-2@1&2&-1)|=10i+j+12k
Notes
1)AX(BXC)=(A.C)B-(A.B)C
2)(AXB)X(CXD)=(A.CXD)B-(B.CXD)A
Home Work:-
A)Find parametric equations for the lines(1-6)?
B) Find equations for the planes the following exercises?
1)
2)
3)
4)
5)
6)
C) find the distance from the point to the line.
1)
2)
3)
4)
D)Find the angles between the planes in Exercises (1,2)?
1)
2)
CHAPTERII
VECTOR –VALUED FUNCTION AND MOTION IN SPACE
WHEN a particle moves through space as a function of time each of its rectangular coordinates is a function of time we may describe the particle path with a triple equation
x=f(t) ,y=g(t) ,z=h(t)
The particle position vector (r(t) ) ?=f(t)i+g(t)j+h(t)k
1)LIMITS
IF L=L1i+L2j+L3k is a vector in space
F(t) is a vector function
F(t)=f(t)i+g(t)j+h(t)k
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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