Page 1 - XII-CH10-VECTOR-LESSON NOTES

XII-CH10-VECTOR

LESSON NOTES

Basic Concepts & Formulae :

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1. If a and b are the position vector of two points A and B, then AB b a .
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2. If a a i b j c k 1  1  1 , then |a| a  b  c .
2
2
2
1
1
1

   a
3. If a is any given vector then unit vector in the direction of a , i.e. a  .
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|a|
  
4. (i) Collinearity of three points: Three points with position vectors, a, b,c are collinear iff there
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  
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exist scalars x, y, z not all zero such that x. a y.b z.c  0 , where x y z 0 .
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(ii) Coplanarity of three vectors: Let a and b be two given non-zero non-collinear vectors.
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Then any vector r coplanar with a and b can be uniquely expressed as r  x a y b for some
scalars x and y.
5. Section formula: The position vector of a point C which is collinear with points A and B
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
  AC m n a mb
and whose position vectors are a , b such that  is .

CB n m n
6. Two vectors are said to be orthogonal if they are perpendicular to each other.
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
7. The dot product (scalar product) of two vectors a and b is given by a.b |a||b|cos , where
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 is the angle between a and b .
8. Properties of dot product.
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(i) a.b  b. a (commutativity)
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
(ii) a.(  b) (a). b (a. b), is a scalar.
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(iii) a.(b  c) a. b a. c (distributive property)
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
     

(iv) a. b 0  a 0, b 0or a 


b
         

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
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(v) If a a i a j a k and b b i b j b k , then a. b a b 1 1  a b  a b .

2 2
3 3
3
2
1
3
2
1
 
  a . b
(vi) Projection of a onb  and
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|b|
 
  a . b
Projection of b on a 

|a|