Math Problem Statement

Statement:  The sum of the magnitudes of two vectors is the same as the magnitude of the sum of the two vectors.  That is, ||a|| + ||b|| = ||a + b||. 

Explain when the statement is valid and give an example of when it is not valid.

Solution

The statement a+b=a+b||\mathbf{a}|| + ||\mathbf{b}|| = ||\mathbf{a} + \mathbf{b}|| is valid only in certain specific cases, namely when the two vectors a\mathbf{a} and b\mathbf{b} point in exactly the same direction or when one or both vectors have zero magnitude. Let us analyze this statement and its conditions for validity in detail:


When the Statement is Valid

  1. Vectors Point in the Same Direction:
    If a\mathbf{a} and b\mathbf{b} are parallel and point in the same direction, the magnitudes simply add up. In this case: a+b=a+b.||\mathbf{a}|| + ||\mathbf{b}|| = ||\mathbf{a} + \mathbf{b}||. This happens because a+b\mathbf{a} + \mathbf{b} results in a vector with the combined magnitude of both vectors, as there is no "angle" between them to reduce the resultant magnitude.

    Example:
    Let a=3i\mathbf{a} = 3\mathbf{i} and b=4i\mathbf{b} = 4\mathbf{i}.
    a=3,b=4,a+b=(3+4)i=7.||\mathbf{a}|| = 3, \quad ||\mathbf{b}|| = 4, \quad ||\mathbf{a} + \mathbf{b}|| = ||(3+4)\mathbf{i}|| = 7. Thus: 3+4=7.3 + 4 = 7.

  2. One or Both Vectors are Zero:
    If a=0\mathbf{a} = \mathbf{0} or b=0\mathbf{b} = \mathbf{0}, the equality holds trivially because adding zero to a vector does not change its magnitude.

    Example:
    Let a=0\mathbf{a} = 0 and b=5j\mathbf{b} = 5\mathbf{j}.
    a=0,b=5,a+b=b=5.||\mathbf{a}|| = 0, \quad ||\mathbf{b}|| = 5, \quad ||\mathbf{a} + \mathbf{b}|| = ||\mathbf{b}|| = 5. Thus: 0+5=5.0 + 5 = 5.


When the Statement is Not Valid

The statement is not valid when a\mathbf{a} and b\mathbf{b} are not parallel and point in different directions. In such cases, the magnitude of their sum is generally less than the sum of their magnitudes, because the resultant vector is shorter due to the angle between them.

Reason:
When a\mathbf{a} and b\mathbf{b} are not parallel, the triangle inequality for vector magnitudes applies: a+ba+b,||\mathbf{a} + \mathbf{b}|| \leq ||\mathbf{a}|| + ||\mathbf{b}||, and equality holds only when a\mathbf{a} and b\mathbf{b} are in the same direction.

Example:
Let a=3i\mathbf{a} = 3\mathbf{i} and b=4j\mathbf{b} = 4\mathbf{j}.
Here, a\mathbf{a} and b\mathbf{b} are perpendicular.
a=3,b=4,a+b=3i+4j=(3,4)=32+42=5.||\mathbf{a}|| = 3, \quad ||\mathbf{b}|| = 4, \quad ||\mathbf{a} + \mathbf{b}|| = ||3\mathbf{i} + 4\mathbf{j}|| = ||(3, 4)|| = \sqrt{3^2 + 4^2} = 5. Thus: a+b=3+4=7,a+b=5.||\mathbf{a}|| + ||\mathbf{b}|| = 3 + 4 = 7, \quad ||\mathbf{a} + \mathbf{b}|| = 5. In this case: 3+45.3 + 4 \neq 5.


Summary

  • The statement is valid when a\mathbf{a} and b\mathbf{b} point in the same direction or one is zero.
  • It is not valid when a\mathbf{a} and b\mathbf{b} are not parallel or point in different directions.

Questions for Further Exploration

  1. How does the angle between two vectors affect the magnitude of their sum?
  2. Can a+b||\mathbf{a} + \mathbf{b}|| ever be greater than a+b||\mathbf{a}|| + ||\mathbf{b}||?
  3. What role does the dot product play in determining the validity of the given statement?
  4. How can we geometrically represent the relationship between a\mathbf{a} and b\mathbf{b} in this context?
  5. Does this statement have any relevance in physics, such as in force or velocity addition?

Tip: Always visualize vectors geometrically when reasoning about their sums and magnitudes—drawing diagrams helps clarify when the equality holds.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Vector Algebra
Vector Magnitudes
Triangle Inequality

Formulas

||a|| + ||b|| = ||a + b|| (Equality Condition)
||a + b|| ≤ ||a|| + ||b|| (Triangle Inequality)

Theorems

Triangle Inequality for Vectors

Suitable Grade Level

Grades 10-12