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Properties of Scalar Product

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Properties of Scalar Product - Lesson Summary

Dot product of vectors   a→ and b→:

a→ . b→ =  |a→| |b→|  cos ϴ

a→ . (b→ + c→) = a→ . b→ + a→ . c→


Distributive property of dot product over vector addition:

Multiplication of dot product of vectors with a scalar:

Given a scalar λ:

λ (a→ . b→) = (λ a→). b→ = a→ .(λ b→)

Let a→ = a1 i ^ + a2  j ^ + a3  k ^  

b→ = b1 i ^ + b2  j ^ + b3  k ^

a→ . b→ = (a1 i ^ + a2  j ^ + a3  k ^ ).(b1 i ^ + b2  j ^ + b3  k ^ )

⇒ a→ . b→ = 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  

 ⇒ a→ . b→ = a1b1 (1) + a1b2 (0) + a1b3 (0) + a2b1 (0) + a2b2 (1) + a2b3 (0) + a3b1 (0) + a3b2 (0) + a3b3 (1)

⇒ a→ . b→ = a1b1 + a2b2  + a3b3


Projection of a Vector

In ∆ OBA:

cos θ = OB OA     

⇒ OB = OA cos θ        

⇒ OB → = OA → cos θ

⇒ p→ = a→  cos θ

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

|p→| = |a→| cos ϴ

If ϴ = 0o:

p→ = a→ cos 0°

p→=a→ (since cos 0° = 1)


If 0o < ϴ < 90o, then the direction of p→ is the same as the direction of line l.

p→=a→ cos θ

If ϴ = 90o:

p → = a→ cos 90°

⇒ p→ =O→ (since cos 90° = 0)

If 90o < ϴ < 180o, then the direction of p→ is opposite to the direction of line l.

If ϴ = 180o:

p→ =    a→ cos 180°

⇒ p→   = - a→ (Since cos 180° = -1 )

If 180o < ϴ < 270o, then the direction of p→ is opposite to the direction of line l.

If ϴ = 270o:

p→ = a→ cos 270°

⇒ p→ = O→ ( Since cos 270° = 0)

If 270o < ϴ < 360o, then the direction of p→ is the same as the direction of line l.

Observation:

If p ^ is a unit vector along line l:

p→ = a→ . p ^

The projection of vector a→ along another vector b→ is given by:

p→ = a→ .  b ^

where  b ^ is a unit vector along b→

Since  b ^ =   b → | b → |     , we have:

p→ = a→ .   b → | b → |

Or p→ =   1 | b → | (a→ . b→)

The component form of vector

a→ = a1 i ^ + a2  j ^ + a3  k ^  :

a1 = Projection of a→ on the X-axis

a2 = Projection of a→ on the Y-axis

a3 = Projection of a→ on the Z-axis

Direction cosines of vector a→ = a1 i ^ + a2  j ^ + a3  k ^   are given by:

cos α =  a 1 a →  

cos β =  a 2 a →    

cos γ =  a 3 a →  

If a→ is a unit vector: a= cos α  i ^ + cos β  j ^ + cos ɣ  k ^   :

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