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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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