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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)

· . = a^{2}

· Projection of on the line of = .

[

a

]

b

2. Distributive law

.( + ) = . + .

3. Angle between two vectors

a)

If q is the angle between the vectors = a_{1}i + a_{2}j + a_{3}k and = b_{1}i + b_{2}j + b_{3}k Then, a = and b = Now, . = ab cos q and . = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}

\ cos q =

.

ab

=

a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}

b)

If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the direction ratios of two lines and if q is the angle between them, then

cos q =

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}

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

a_{1}

a_{2}

a_{3}

b_{1}

b_{2}

b_{3}

5. Scalar triple product

Scalar triple product of the vectors , , .

. ´ =

c_{1}

c_{2}

c_{3}

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 = . ´