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summary of this topic

Properties of Dot Product

Properties of Dot Product

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 »  vector dot product is bilinear
    →  `vec p cdot (lambda vec q + vec r) ` ` = lambda (vec p cdot vec q) ` ` + vec p cdot vec r`

Bilinear Property

plain and simple summary

nub

plain and simple summary

nub

dummy

 •  Dot product is bilinear.

simple steps to build the foundation

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simple steps to build the foundation

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In this page, you will learn about the bilinear property of vector dot product.


Keep tapping on the content to continue learning.
Starting on learning "Bilinear Property of vector dot product". ;;

Dot product is defined between `vec p, vec q in bbb V` and the result `vec p cdot vec q in RR`. This can be considered as a transformation. What type of transformation is this?

  • unary
  • binary
  • ternary

The answer is 'binary', as the input is two vectors. 'bi' means two.

Which of the following property is true for Bilinear transformation?

  • `T(x+y, z)=T(x,z)+T(y,z)`
  • `T(ax,z)=aT(x,z)`
  • `T(ax+y, z)= aT(x,z)+T(y,z)`
  • All the above

The answer is 'All the above'.

This property defines linearity for binary operator `T`. That is why it is named Bilinear.

Consider the dot product as a binary transform `T(vec p, vec q) = vec p cdot vec q`. Is dot product bilinear?

  • Yes
  • No

The answer is 'Yes'. The dot product is distributive over vector addition and also satisfies the scalar multiple property.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Bilinear Property: For any vector `vec p, vec q, vec r in bbb V` and `lambda in RR`
`(lambda vec p + vec q) cdot vec r = lambda (vec p cdot vec r) + (vec q cdot vec r)`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Given `vec x cdot vec z = 2` and `vec y cdot vec z = 1`, what is `(3 vec x + 2 vec y) cdot vec z`?

  • `5`
  • `6`
  • `8`
  • `9`

The answer is '`8`'
`(3 vec x + 2 vec y) cdot vec z`
`quad = 3 vec x cdot vec z + 2 vec y cdot vec z`

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Progress

Dot product is defined between vector p and q in vector space v. And the result vector p dot vector q in real numbers. This can be considered as a transformation. What type of transformation is this?
unary
unary
binary
binary
ternary
ternary
The answer is "binary", as the input is two vectors. bi means two.
Which of the following property is true for Bilinear transformation?
1
2
3
4
The answer is "All the above". This property defines linearity for a binary operator T. That is why it is names bilinear.
Consider the dot product as a binary transform : t of vector p, vector q =, vector p dot vector q. Is dot product bilinear?
yes;s
Yes
no
No
The answer is 'Yes'. The dot product is distributive over vector addition and also satisfies the scalar multiple property.
Dot product is bilinear.
Bilinear Property: For any vector p, q, r in vector space v, and lambda in real numbers. lambda times vector p plus vector q, dot vector r =, lambda times vector p dot vector r ,+, vector q dot vector r.
Given vector x dot vector z = 2 and vector y dot vector z = 1, what is 3 times vector x plus 2 times vector y, dot vector z?
1
2
3
4
The answer is "8".

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