Important Formulas: Vector Algebra

 Page 1


Page # 32
VECTORS
?. Position Vector Of A Point:
let O be a fixed origin, then the position vector of a point P is the vector
OP . If a
?
 and b
?
 are position vectors of two points A and B, then,
AB = a b
?
?
? = pv of B ? pv of A. A.
DISTANCE FORMULA : Distance between the two points A ) a (
?
 and B ) b (
?
is AB = b a
?
?
?
SECTION FORMULA : 
n m
b m a n
r
?
?
?
?
?
?
. Mid point of AB =
2
b a
?
?
?
.
??. Scalar Product Of Two Vectors: a . b
?
= |
a
?
| | 
b
?
| cos ?, where |
a
?
|, | 
b
?
|
are magnitude of a
?
 and b
?
 respectively and ? is angle between a
?
 and b
?
.
1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of 
| b |
a.b
a on b ?
?
?
?
?
?
2. If 
?
a = a
1
i + a
2
j + a
3
k & b
?
 = b
1
i + b
2
j + b
3
k then a.b
?
?
 = a
1
b
1 
+ a
2
b
2 
+ a
3
b
3
3. The angle ? between a&b
?
?
 is given by  
|a || |b
a.b
cos ?
?
?
?
? ?
,  0 ?? ?? ?? ?
4. a . b 0 a b
?
?
?
?
? ? ? (a ? 0 b ? 0)
?
?
Page 2


Page # 32
VECTORS
?. Position Vector Of A Point:
let O be a fixed origin, then the position vector of a point P is the vector
OP . If a
?
 and b
?
 are position vectors of two points A and B, then,
AB = a b
?
?
? = pv of B ? pv of A. A.
DISTANCE FORMULA : Distance between the two points A ) a (
?
 and B ) b (
?
is AB = b a
?
?
?
SECTION FORMULA : 
n m
b m a n
r
?
?
?
?
?
?
. Mid point of AB =
2
b a
?
?
?
.
??. Scalar Product Of Two Vectors: a . b
?
= |
a
?
| | 
b
?
| cos ?, where |
a
?
|, | 
b
?
|
are magnitude of a
?
 and b
?
 respectively and ? is angle between a
?
 and b
?
.
1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of 
| b |
a.b
a on b ?
?
?
?
?
?
2. If 
?
a = a
1
i + a
2
j + a
3
k & b
?
 = b
1
i + b
2
j + b
3
k then a.b
?
?
 = a
1
b
1 
+ a
2
b
2 
+ a
3
b
3
3. The angle ? between a&b
?
?
 is given by  
|a || |b
a.b
cos ?
?
?
?
? ?
,  0 ?? ?? ?? ?
4. a . b 0 a b
?
?
?
?
? ? ? (a ? 0 b ? 0)
?
?
Page # 33
?? ?. Vector Product Of Two Vectors:
1. If
?
?
a b & are two vectors & ? is the angle between them then
n sin b a b x a
?
?
?
?
?
? ?
,
where
?
n is the unit vector perpendicular to both
?
?
a b & such that n & b , a
?
?
?
forms a right handed screw system.
2. Geometrically
b x a
?
?
 = area of the parallelogram whose two adjacent sides
are represented by b & a
?
?
.
3. 0 k
ˆ
k
ˆ
j
ˆ
j
ˆ
i
ˆ
i
ˆ
?
? ? ? ? ? ? ; j
ˆ
i
ˆ
k
ˆ
, i
ˆ
k
ˆ
j
ˆ
, k
ˆ
j
ˆ
i
ˆ
?? ? ? ??
4. If
?
a = a
1
i
ˆ
 +a
2
j
ˆ
 + a
3
k
ˆ
 &
?
b = b
1
i
ˆ
 + b
2
j
ˆ
 + b
3
k
ˆ
 then
3 2 1
3 2 1
b b b
a a a
k
ˆ
j
ˆ
i
ˆ
b a ? ?
?
?
5. b and a o b a
?
? ?
?
?
? ? ? are parallel (collinear)
)0b,0a( ? ?
?
?
i.e. b K a
?
?
? , where K is a scalar. .
6. Unit vector perpendicular to the plane of 
a x b
a x b
n
ˆ
a & b is ?
?
?
?
?
?
? ?
? If a,b&c
?
?
?
 are the pv’s of 3 points A, B & C then the vector area of triangle
ABC = ? b xb c cxa ? xa
2
1
? ? ?
? ?
?
? ? .
The points A, B & C are collinear if bx bxc c ax 0 a
?
? ? ?
? ?
?
???
? Area of any quadrilateral whose diagonal vectors are
? ?
d
1
& d
2
 is given by
1 2
d xd
2
1
? ?
? Lagrange's Identity : 
.ab .bb
aa. ab.
(ax )b a b (a. )b
2
2
2
2
? ? ?
?
?
? ? ?
?
?
?
?
?
?
? ? ?
Page 3


Page # 32
VECTORS
?. Position Vector Of A Point:
let O be a fixed origin, then the position vector of a point P is the vector
OP . If a
?
 and b
?
 are position vectors of two points A and B, then,
AB = a b
?
?
? = pv of B ? pv of A. A.
DISTANCE FORMULA : Distance between the two points A ) a (
?
 and B ) b (
?
is AB = b a
?
?
?
SECTION FORMULA : 
n m
b m a n
r
?
?
?
?
?
?
. Mid point of AB =
2
b a
?
?
?
.
??. Scalar Product Of Two Vectors: a . b
?
= |
a
?
| | 
b
?
| cos ?, where |
a
?
|, | 
b
?
|
are magnitude of a
?
 and b
?
 respectively and ? is angle between a
?
 and b
?
.
1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of 
| b |
a.b
a on b ?
?
?
?
?
?
2. If 
?
a = a
1
i + a
2
j + a
3
k & b
?
 = b
1
i + b
2
j + b
3
k then a.b
?
?
 = a
1
b
1 
+ a
2
b
2 
+ a
3
b
3
3. The angle ? between a&b
?
?
 is given by  
|a || |b
a.b
cos ?
?
?
?
? ?
,  0 ?? ?? ?? ?
4. a . b 0 a b
?
?
?
?
? ? ? (a ? 0 b ? 0)
?
?
Page # 33
?? ?. Vector Product Of Two Vectors:
1. If
?
?
a b & are two vectors & ? is the angle between them then
n sin b a b x a
?
?
?
?
?
? ?
,
where
?
n is the unit vector perpendicular to both
?
?
a b & such that n & b , a
?
?
?
forms a right handed screw system.
2. Geometrically
b x a
?
?
 = area of the parallelogram whose two adjacent sides
are represented by b & a
?
?
.
3. 0 k
ˆ
k
ˆ
j
ˆ
j
ˆ
i
ˆ
i
ˆ
?
? ? ? ? ? ? ; j
ˆ
i
ˆ
k
ˆ
, i
ˆ
k
ˆ
j
ˆ
, k
ˆ
j
ˆ
i
ˆ
?? ? ? ??
4. If
?
a = a
1
i
ˆ
 +a
2
j
ˆ
 + a
3
k
ˆ
 &
?
b = b
1
i
ˆ
 + b
2
j
ˆ
 + b
3
k
ˆ
 then
3 2 1
3 2 1
b b b
a a a
k
ˆ
j
ˆ
i
ˆ
b a ? ?
?
?
5. b and a o b a
?
? ?
?
?
? ? ? are parallel (collinear)
)0b,0a( ? ?
?
?
i.e. b K a
?
?
? , where K is a scalar. .
6. Unit vector perpendicular to the plane of 
a x b
a x b
n
ˆ
a & b is ?
?
?
?
?
?
? ?
? If a,b&c
?
?
?
 are the pv’s of 3 points A, B & C then the vector area of triangle
ABC = ? b xb c cxa ? xa
2
1
? ? ?
? ?
?
? ? .
The points A, B & C are collinear if bx bxc c ax 0 a
?
? ? ?
? ?
?
???
? Area of any quadrilateral whose diagonal vectors are
? ?
d
1
& d
2
 is given by
1 2
d xd
2
1
? ?
? Lagrange's Identity : 
.ab .bb
aa. ab.
(ax )b a b (a. )b
2
2
2
2
? ? ?
?
?
? ? ?
?
?
?
?
?
?
? ? ?
Page # 34
?V. Scalar Triple Product:
? The scalar triple product of three vectors
?
?
?
a b c , &
 is defined as:
?
?
? ?
?
?
a x b c a b c . ?
sin cos ? ? .
? Volume of tetrahydron ] c b a [ V
?
?
?
?
? In a scalar triple product the position of dot & cross can be interchanged
i.e.
?
?
? ?
?
? ?
?
?
?
? ? ? ?
?
abxc axbcORabc bca cab .( )( ). [] [] [] ? ? ?
?
?
?
? ? ?
?
?
?
? ? ?
?
a bx c a cx b i e a b c a c b . ( ) .( ) . . [ ] [ ] ? ? ? ?
? If 
?
a = a
1
i+a
2
j+a
3
k;
?
b = b
1
i+b
2
j+b
3
k &
?
c = c
1
i+c
2
j+c
3
k then
3 2 1
3 2 1
3 2 1
c c c
b b b
a a a
] c b a [ ?
?
?
?
.
? If
?
?
?
a b c , , are coplanar 0 ] c b a [ ? ?
?
?
?
.
? Volume of  tetrahedron OABC with O as origin & A( a
?
),  B( b
?
) and C( c
?
)
be the vertices = 
] c b a [
6
1
?
?
?
? if the pv’s of its vertices
are
?
?
?
?
a , b , c & d are given by 
1
4
[ ]
?
?
?
?
a ? b ? c ? d .
V. Vector Triple Product:
?
?
?
a x ( b x c) = (. ) (. )
? ?
?
?
?
?
a c b ? a b c , ( )
?
?
?
a x b x c =( .) ( .)
? ?
? ?
? ?
a c b ? b c a
? ( ) ( )
?
?
? ?
?
?
a xb xc ? a x b xc , in general