Inequalities Based on Dot Product
There are various inequalities based on the dot product of vectors, such as:
- Cauchy – Schwartz inequality
- Triangle Inequality
Let’s discuss these in detail as follows:
Cauchy – Schwartz inequality
According to this principle, for any two vectors a and b, the magnitude of the dot product is always less than or equal to the product of magnitudes of vector a and vector b
|a.b| ≤ |a| |b|
Proof:
Since, a.b = |a| |b| cos α
We know that 0 < cos α < 1
So, we conclude that |a.b| ≤ |a| |b|
Triangle Inequality
For any two vectors a and b, we always have
|a+ b| ≤ |a| + | b|
Proof:
|a+b|2=|a+b||a+b|
= a.a + a.b +b.a+ b.b
= |a|2+ 2a.b+|b|2 (dot product is commutative)
≤ |a|2 + 2|a||b| + |b|2
≤ (|a| + |b|)2
This proves that |a + b| ≤ |a| + |b|
Dot and Cross Products on Vectors
A quantity that is characterized not only by magnitude but also by its direction, is called a vector. Velocity, force, acceleration, momentum, etc. are vectors.
Vectors can be multiplied in two ways:
- Scalar product or Dot product
- Vector Product or Cross product
Table of Content
- Scalar Product/Dot Product of Vectors
- Projection of one vector on other Vector
- Properties of Scalar Product
- Inequalities Based on Dot Product
- Cross Product/Vector Product of Vectors
- Properties of Cross Product
- Cross product in Determinant Form
- Dot and Cross Product
- FAQs on Dot and Cross Products on Vectors
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