F6 Mathematics T
1/5
Revision Notes on Chapter 6 : Vectors (Term 1) Name : ______________________________ _________________________ _____
Date : __________________
6.1 : VECTORS IN 2 & 3-D (A) : Unit Vectors & Position Vectors
x y 1). Position vector of a point A( x, y , z ) r OA xi yj zk z 2). Length of OA
r
x2 y 2 z2
3). Unit vector in the direction of OA (with length of 1 unit)
OA
r
r
OA
(B) : Algebraic Properties of Vectors
x2 x1 AB OB OA y2 y1 z z 2 1
1). If point A ( x1 , y1 , z1 ) and B ( x2 , y2 , z 2 ) ,
2). Distance between point A( x1 , y1, z 1 ) and B( x2 , y2 , z2 ) = OC
3).
a b
λ
A
C
μ
a
x1 x2
2
y1 y2
2
2
z1 z 2
B b
O 4). At x-axis, y = 0, z = 0 ; At y-axis, x = 0, z = 0 ; At z-axis, x = 0, y = 0 (C) : Scalar Product ( Dot Product ) of 2 Vectors
a.b
1).
a
b
cos
;
cos
a b
a
+ve or – or – ve ve scalar
Ө
b
a1 b1 a.b a2 . b2 a1b1 a 2b2 a 3b3 a b 3 3
2).
i .i j. j k .k 1 ; i . j j.k k .i 0
3). 4).
a.b b.a
5).
a.(b c) a.b a. c
6). If
a.b
k
;
if a .b 0 a b and vice versa
example : a . b a . is a constant, then a . k b k ( a . b ) , for example
b b
1 b
(a .b )
ab
(D) : Vector Product ( Cross Product ) of 2 Vectors
1).
a b=( a
2/5
b sin ) n
b
ˆ
The direction can be determined by Right-hand-rule.
2).
a b
i
j
k
a1
a2
a3
b1
b2
b3
( a2b3 a3b2 ) i ( a1b3 a3b1) j ( a1b2 a2b1) k z
3). i i j j k k 0 ; i j k , j k i , k i j ; j i - k , k j - i , i k - j
k i
i
If a b 0 , then a // b and vice versa 4).
a b - b a
5).
a (b c) a b a c
6).
If
k
x
is a constant, then a k b k ( a b )
7). Area of triangle ABC
1 2
AB AC
8). Area of parallelogram ABCD
AB AD
(E) : Application of Dot & Cross Product of Vectors
1).
Volume of a cuboid
a b c ( a b sin 90) c a b c cos 0 ) a b ( a b a b.c 2).
a
Ө
c b
a
Projections :
C
AC
AC b
ˆ
A
E
AC . b
ˆ
b
D
j
i
y
6.2 : Vector Geometry
3/5
(A) : Line (a point A ( x1 , y1 , z 1 ) on the line and a direction vector b ai bj ck are needed)
1). Vector Equation of a line, l : r a t b ( for example : r ( 3i 2 j k ) t( 3i - j 8k ) )
x y z
x1 y 1 z 1
a b t c
Vector from the origin
Position vector
Direction vector of the line
pointing to the line
of a point on the line
(e.g. AB OB - OA , simplified by factorization, where t is just a constant )
2). Cartesian Equation of a line : x - x1 y - y1
a
b
z - z1
x 1
( for example :
c
1
y2
1
z 3
3
)
Must know : i ). interchange both vector & Cartesian equations of a line. ii ). given 2 points, form equation. iii). show a given point is on a given line. iv). direction vector of x-axis or parallel to x -axis = i , direction vector of y-axis or parallel to y-axis = j , direction vector of z-axis or parallel to z-axis = k 3).Shortest distance (perpendicular distance) from a point C to a line = p
AC
b
ˆ
Must know : i ). given 1 point & 1 line, find p. (hint: using p
AC
ii ). given 2 skewed lines, find p. (hint: using p
AC . n
iii). given 2 parallel lines, find p. (hint: using p
AC
b ) ˆ
, where n
ˆ
ˆ
b1 b2 b1 b2
)
b ) ˆ
iv). always use modulus sign when finding distance using dot product of vectors. (B) : Plane (a point A ( x1 , y1 , z 1 ) on the plane and a normal vector n ai bj ck are needed)
1). Vector Equation of a plane, :
x y z
.
Vector from the origin Pointing to the plane
r.n d
a b c
( for example : r . ( 2 i - 2 j k ) - 3 )
Normal vector to the plane
x1a y1b z1c
unique scalar for a specific plane
2). Cartesian Equation of a plane : ax by cz d
where d OA . n
4/5
Must know : i ). interchange both vector & Cartesian equations of a plane. ii ). given 1 point on a plane & the normal vector, find the equation of plane. iii). find the intersection point between a plane and x, y or z axis. (e.g. for x-axis, y=0,z=0; substitute these into the plane’s Cartesian eq. to find x, i.e. (x,0,0)) iv). given 3 points on a plane,find the equation of the plane. v ). given 2 points on a plane & a vector on the plane or a parallel line equation, find the equation of the plane. vi). given 1 point on a plane & 1 line equation on the plane, find the equation of plane. vii).show a given point is on a given plane. viii).show a given line is on a given plane.(hint: substitute the r of the line into the r of the plane and if the dot product is equal to the value of d, then the line is on the plane) ix). given 1 point, A above plane & 1 plane, find the perpendicular intersecting point, B from A to the plane. (hint: find OB OA t n , then substitute the co-ordinates of B into the plane equation to find t ) x ). Given a line, l 1 on a plane,
1
which is perpendicular to another plane 2 , find
1
.
(hint: n1 of 1 n2 b1 & obtain a point from l 1) 3). Shortest distance (perpendicular distance) from the origin to a plan e = p
r.n ˆ
4). Shortest distance from a point ( x1 , y1 , z1 ) to the plane ( ax by cz d 0 )
ax1 by1 cz1 d a 2 b2 c 2
(note: this formula is derived from p p2 p1 and can be used directly) (C) : Angles (All formulae in this part are derived using the dot product of 2 vectors) (if cos is – ve, find also the acute angle of )
1). Angle between 2 lines = , (can be acute or obtuse), where b1 . b2 cos b1 b2 2). Angle between a line & a plane = , where cos sin
and
sin
n b
b.n b
3). Angle between 2 planes = , where cos
n2
n1 . n2 n1 n2
plane
n
plane 1
n1
plane 2
5/5
(D) : Intersection
1). To find the intersecting point between lines l 1: r 1 a1 b1 & l 2 : r 2
a2
b2 ,
Steps: i ). Start with a 1 b1 = a2 b2 ii ). Then equating the 3 coefficient of i , j , k respectively to find the values of and . (note: the values of and must satisfy the 3 equations or otherwise the 2 lines not intersecting) iii). Substitute the values of and back into l 1 or l 2 to find the intersecting point. 2). To find the intersecting point between a line l, r a tb and a plane , r . n d , Steps: i ). Substitute the r of the line into the r of the plane to find the value of t . ii ). Substitute the value of t back into the line equation to find the intersecting point. 3). To find the intersecting line between the plane 1 and 2 , Steps: i ). The direction vector of the intersecting line = n1 n2 ii ). Find a point on the intersecting line by using Cartesian equations of 1 and 2 : a). eliminate the variable z to obtain 1 equation with only x and y variables. b). let x = any number which will cause y to be an integer. c). substitute the values of x and y into one of the planes’ Cartesian equation to find z value. d). write the intersecting line equation as r a tb
@@@@@@@@@@@@@@@@@@@@@@@
