NHAA/IMK/UNIMAP
VECTORS
DOT PRODUCT
CROSS PRODUCT
APPLICATIONS
NHAA/IMK/UNIMAP
DIRECTIONS OF ANGLES & DIRECTIONS OF COSINES
,, - Are the angles that the vector OP makes with positive axis- Knows as the direction angles of vector OP
DIRECTION OF COSINES
180,,0;cos
cos
cos
OP
z
OP
y
OP
x
NHAA/IMK/UNIMAP
Example 1Find the direction cosines and direction angles of:
3,2,11,1,0
3,1,2
QPii
ui
NHAA/IMK/UNIMAP
DOT PRODUCT Also known as inner product or scalar product The result is a scalar
If and then:321 ,, uuuu 321 ,, vvvv
332211
321321
,,.,,.
vuvuvu
vvvuuuvu
NHAA/IMK/UNIMAP
Example 2If and :
2 9
i u v
ii u v u
1,3,2 u 3,2,0 v
NHAA/IMK/UNIMAP
DOT PRODUCT Angle Between 2 Vectors
u
v vu
vu.cos
If the vectors lies on the same line or parallel to each other, then
0u v u
v
NHAA/IMK/UNIMAP
Example 3Find the angles between and
6,0,43,0,2
4,3,23,2,1
vuii
vui
u v
NHAA/IMK/UNIMAP
DOT PRODUCTProperties of Dot Product
NHAA/IMK/UNIMAP
CROSS PRODUCT The result is a vector
If and then: (determinant of the matrix)
321 ,, uuuu 321 ,, vvvv
kvuvujvuvuivuvu
vvv
uuu
kji
vu
122113312332
321
321
NHAA/IMK/UNIMAP
Example 3Find the cross product between and
6,0,43,0,2
4,3,23,2,1
vuii
vui
u v
NHAA/IMK/UNIMAP
CROSS PRODUCTProperties of Cross Product jkiijkkijix
jikikjkjiviii
kkjjiivii
vuvuvi
vuv
auvi
vkuvukvukiii
wuvuwvuii
uvvui
,,
,,
0
sin
0
000
if u and v are parallel
NHAA/IMK/UNIMAP
APPLICATIONS1. PROJECTIONSThe vector projection of u = onto a nonzero vector Is the vector determined by dropping a perpendicular from Q to the line PS.
PQ PSv
PR
u
v
Q
P R S
uprojv
NHAA/IMK/UNIMAP
APPLICATIONS1. PROJECTIONS• Scalar projection• The magnitude of the vector projection:
• Vector projection
v
vuucompv
.
v
vucompuproj vv
NHAA/IMK/UNIMAP
APPLICATIONS1. AREA OF TRIANGLE & PARALLELOGRAMThe magnitude of uxv is the area of parallelogram
u
vsinvh
vu
vu
heightbase
sin
ramparallelog of area
NHAA/IMK/UNIMAP
APPLICATIONS1. AREA OF TRIANGLE Half of the area of parallelogram
vu
vu
2
1
sin2
1
ramparallelog of area2
1 triangleof area
v
u
NHAA/IMK/UNIMAP
APPLICATIONS
i. Parametric Equations of a Line in
ii. Equations of Planes
iii. Distance from a Point to the Plane
3R
NHAA/IMK/UNIMAP
APPLICATIONSiii. Lines & Line Segment in SpaceParametric Equations
z
y
x
P0(x0,y0,z0) LP(x,y,z)
v
Line L is the set of all points P(x,y,z) for which parallel to :
PP0 v
RtvtPP ,0
NHAA/IMK/UNIMAP
APPLICATIONSTherefore the parametric equation for L :
ctzzctzz
btyybtyy
atxxatxx
kcjbiatkzzjyyixx
vtPP
00
00
00
000
0
ParametricEquation
NHAA/IMK/UNIMAP
Cartesian equation:
c
zz
b
yy
a
xxt 000
NHAA/IMK/UNIMAP
Example 3:Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to 2,4,2 v
NHAA/IMK/UNIMAP
Example 4:Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4)
NHAA/IMK/UNIMAP
APPLICATIONSDistance from point S to line L
S
P v
sinPS
From the properties of Cross Product
v
vuu
vuvu
sin
sin
Formula of Distance from point S to L
v
vPSD
L
u
NHAA/IMK/UNIMAP
Example 5:Find the distance from the point S(1,1,5) to the line
tztytxL 2,3,1:
NHAA/IMK/UNIMAP
APPLICATIONSLines of Intersection
n1
n2
v
Line of intersection
21 nnv Finding v :
NHAA/IMK/UNIMAP
Example 8:Find a vector parallel to the line of intersection of the planes
522
15263
zyx
zyx
NHAA/IMK/UNIMAP
Example 9:Find the parametric equation for the line in which the planes
Intersect.
522
15263
zyx
zyx
NHAA/IMK/UNIMAP
Example 9:Find the point where the line
Intersects the plane
tztytx 1,2,23
8
6623 zyx
NHAA/IMK/UNIMAP
APPLICATIONSEquation of Planes
P0(x0,y0,z0)P(x,y,z)
n
Vector is on the plane M and vector which is perpendicular to M known as normal vector, n
PP0
From the properties of Dot Product
0.0 nPP
NHAA/IMK/UNIMAP
APPLICATIONSEquation of Planes vun
Normal vector n :
u
v
vun
NHAA/IMK/UNIMAP
APPLICATIONSLet and 0000 ,,,,, zyxPcban zyxP ,,
dczbyax
czbyaxczbyax
cbazzyyxx
nPP
000
000
0
0,,.,,
0.
EQUATION OF PLANE
NHAA/IMK/UNIMAP
Example 6:Find an equation of plane through P0(-3,0,7) perpendicular to 1,2,5 n
NHAA/IMK/UNIMAP
Example 7:Find equation of plane through 3 points:
0,3,00,0,21,0,0 CBA
NHAA/IMK/UNIMAP
APPLICATIONSDistance from a Point to the Plane
P0
P
n D
n
nPPD
.0
NHAA/IMK/UNIMAP
APPLICATIONS
222
000
0
,,
,,,,
.
cba
dczbyax
cba
cbazzyyxx
n
nPPD
Equation of plane,With n = <a,b,c>
NHAA/IMK/UNIMAP
Example 10:Find the distance from S(1,1,3) to the plane 6623 zyx