The Relationship Between Vector A Vector B and |A| |B|

This article explores the relationship between the vector sum (vec{a} vec{b}) and the magnitude sum (|vec{a}| |vec{b}|). We will examine conditions under which (vec{a} vec{b}) is less than (|vec{a}| |vec{b}|), and the factors that influence this relationship.

Introduction to Vector Addition and Magnitude

In vector algebra, vector addition is a fundamental operation used to combine two or more vectors into a single vector. It is important to note that vector addition is not commutative. This means that the order in which vectors are added matters. Specifically, (vec{a} vec{b}) is not necessarily equal to (vec{b} vec{a}). Furthermore, the magnitude of the resultant vector is not always the simple sum of the magnitudes of the individual vectors.

Key Concepts: Commutative Property and Magnitude

The commutative property of vector addition does not hold true for vectors, unlike scalar addition. Therefore, while the order of addition might yield the same result, it is not a given. The magnitude of a vector, denoted as (|vec{a}|), represents the length or size of the vector.

When we add two vectors, the magnitude of the resultant vector can be less than, equal to, or greater than the sum of the magnitudes of the vectors. This depends on the direction of the vectors.

Examples and Analysis

Consider the following vectors:

(vec{a} (10, 0, 0)) (vec{b} (0, 1, 0))

The sum of these vectors can be calculated as:

[vec{a} vec{b} (10, 0, 0) (0, 1, 0) (10, 1, 0)]

The magnitude of (vec{a} vec{b}) is:

[|vec{a} vec{b}| sqrt{10^2 1^2 0^2} sqrt{101}]

The magnitude of (vec{a}) is:

[|vec{a}| 10]

The magnitude of (vec{b}) is:

[|vec{b}| 1]

Therefore, (|vec{a}| |vec{b}| 10 1 11).

We can observe that:

[|vec{a} vec{b}| sqrt{101} This demonstrates that the magnitude of the sum of two vectors can indeed be less than the sum of their individual magnitudes.

Conditions When the Sum is Less Than the Magnitude Sum

The vector sum (vec{a} vec{b}) will have a magnitude that is less than (|vec{a}| |vec{b}|) under certain conditions. This typically occurs when the vectors (vec{a}) and (vec{b}) are pointing in opposite directions. In such cases, the vectors will have a resultant vector with a smaller magnitude.

Conclusion

In conclusion, while vector addition is not commutative and the magnitude of the sum of two vectors is not always the simple sum of their magnitudes, there are specific conditions under which the magnitude of the resultant vector can be less than the sum of the magnitudes of the individual vectors. The direction of the vectors plays a crucial role in this relationship.

Related Keywords

Vector Sum

The process of combining two or more vectors into a single vector, resulting in a new vector that represents the combined effect of the original vectors.

Vector Magnitude

The length or size of a vector, denoted as (|vec{a}|), which represents the absolute value of the vector.

Vector Addition

The operation of combining two or more vectors to form a new vector, which is the sum of the individual vectors.

Vector Inequality

The relationship between the resultant vector and the sum of the magnitudes of the individual vectors, which can be less than or equal to it.

Commutative Property

A property of vector addition that states the order in which vectors are added does not affect the result, although in practice, the resultant may have different directions.