MHT-CET : Mathematics Entrance Exam

### MHT - CET : Mathematics - Vectors Formulae Page 3

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1. Scalar product and projection law

 The scalar product of two vectors and , which are inclined at an angle q , is defined as a scalar (i.e. a real number). ab cos q It is denoted by . and hence, is also called the dot product, i.e., . = ab cos q

·  . = .                                                       (commutative property)

·  . = a2

·  Projection of on the line of = .

[

 a

]

·  Projection of on the line of = .

[

 b

]

2. Distributive law

 .( + ) = . + .

3. Angle between two vectors

a)

If q is the angle between the vectors
=
a1i + a2j + a3k and
=
b1i + b2j + b3k
Then,
a =
and
b =
Now, . =
ab cos q and . = a1b1 + a2b2 + a3b3

\ cos q =

 . ab

=

 a1b1 + a2b2 + a3b3 .

b)

If
a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and if q is the angle between them, then

cos
q

 a1a2 + b1b2 + c1c2 .

4. Vector product

Vector product of and is denoted by ´ and is also called the cross product.
´ = (ab sin q) where is the unit vector in the direction perpendicular to the plane of and

·  ´ = - ´

·  sin q

 | ´ | ||.| |

·  ´

 i j k a1 a2 a3 b1 b2 b3

5. Scalar triple product

Scalar triple product of the vectors , , .

. ´

 a1 a2 a3 b1 b2 b3 c1 c2 c3

Note: The scalar triple product , ´ is also denoted by [ ] and is called the box product.

6. Volume of parallelopiped

 The expression for the volume of the parallelopiped whose co-terminus edges are the vectors , , is given as volume of the parallelopiped = . ´

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