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# Properties of Scalar Product

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#### Properties of Scalar Product - Lesson Summary

Dot product of vectors   aâ†’ and bâ†’:

aâ†’ . bâ†’ =  |aâ†’| |bâ†’|  cos Ï´

aâ†’ . (bâ†’ + câ†’) = aâ†’ . bâ†’ + aâ†’ . câ†’

Distributive property of dot product over vector addition:

Multiplication of dot product of vectors with a scalar:

Given a scalar Î»:

Î» (aâ†’ . bâ†’) = (Î» aâ†’). bâ†’ = aâ†’ .(Î» bâ†’)

Let aâ†’ = a1 i ^ + a2  j ^ + a3  k ^

bâ†’ = b1 i ^ + b2  j ^ + b3  k ^

aâ†’ . bâ†’ = (a1 i ^ + a2  j ^ + a3  k ^ ).(b1 i ^ + b2  j ^ + b3  k ^ )

â‡’ aâ†’ . bâ†’ = a1 i ^   . (b1 i ^ + b2  j ^ + b3  k ^ ) +   a2  j ^   . (b1 i ^ + b2  j ^ + b3  k ^ ) + a3  k ^   . (b1 i ^ + b2  j ^ + b3  k ^ )

= a1b1 ( i ^   .  i ^ ) + a1b2 ( i ^   .  j ^ ) + a1b3 ( i ^   .  k ^ ) + a2b1 ( j ^   .  i ^ ) + a2b2 ( j ^   .  j ^ ) + a2b3 ( j ^   .  k ^ ) + a3b1 ( k ^   .  i ^ ) + a3b2 ( k ^   .  j ^ ) + a3b3 ( k ^   .  k ^ )

i ^   .  j ^   =  j ^   .  k ^    =   j ^   .  k ^    = 0

i ^   .  i ^   =  j ^   .  j ^    =   k ^   .  k ^    = 1

â‡’ aâ†’ . bâ†’ = a1b1 (1) + a1b2 (0) + a1b3 (0) + a2b1 (0) + a2b2 (1) + a2b3 (0) + a3b1 (0) + a3b2 (0) + a3b3 (1)

â‡’ aâ†’ . bâ†’ = a1b1 + a2b2  + a3b3

Projection of a Vector

In âˆ† OBA:

cos Î¸ = OB OA

â‡’ OB = OA cos Î¸

â‡’ OB â†’ = OA â†’ cos Î¸

â‡’ pâ†’ = aâ†’  cos Î¸

The magnitude of vector P is called the projection of vector A.

|pâ†’| = |aâ†’| cos Ï´

If Ï´ = 0o:

pâ†’ = aâ†’ cos 0°

pâ†’=aâ†’ (since cos 0° = 1)

If 0o < Ï´ < 90o, then the direction of pâ†’ is the same as the direction of line l.

pâ†’=aâ†’ cos Î¸

If Ï´ = 90o:

p â†’ = aâ†’ cos 90°

â‡’ pâ†’ =Oâ†’ (since cos 90° = 0)

If 90o < Ï´ < 180o, then the direction of pâ†’ is opposite to the direction of line l.

If Ï´ = 180o:

pâ†’ =    aâ†’ cos 180°

â‡’ pâ†’   = - aâ†’ (Since cos 180° = -1 )

If 180o < Ï´ < 270o, then the direction of pâ†’ is opposite to the direction of line l.

If Ï´ = 270o:

pâ†’ = aâ†’ cos 270°

â‡’ pâ†’ = Oâ†’ ( Since cos 270° = 0)

If 270o < Ï´ < 360o, then the direction of pâ†’ is the same as the direction of line l.

Observation:

If p ^ is a unit vector along line l:

pâ†’ = aâ†’ . p ^

The projection of vector aâ†’ along another vector bâ†’ is given by:

pâ†’ = aâ†’ .  b ^

where  b ^ is a unit vector along bâ†’

Since  b ^ =   b â†’ | b â†’ |     , we have:

pâ†’ = aâ†’ .   b â†’ | b â†’ |

Or pâ†’ =   1 | b â†’ | (aâ†’ . bâ†’)

The component form of vector

aâ†’ = a1 i ^ + a2  j ^ + a3  k ^  :

a1 = Projection of aâ†’ on the X-axis

a2 = Projection of aâ†’ on the Y-axis

a3 = Projection of aâ†’ on the Z-axis

Direction cosines of vector aâ†’ = a1 i ^ + a2  j ^ + a3  k ^   are given by:

cos Î± =  a 1 a â†’

cos Î² =  a 2 a â†’

cos Î³ =  a 3 a â†’

If aâ†’ is a unit vector: a= cos Î±  i ^ + cos Î²  j ^ + cos É£  k ^   :