Notes On Properties of Scalar Product - CBSE Class 12 Maths

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

pa  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 ^   :

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

pa  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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