Establish the following vector inequalities geometrically or otherwise: When does the equality sign above apply?
|a+b| <= |a| + |b|
By the triangle law of vector addition, when vectors a and b are placed head to tail, they form a triangle with the resultant a + b as the third side. In any triangle, one side is less than or equal to the sum of the other two sides. Hence, |a+b| |a| + |b|. Equality holds when a and b are in the same direction (collinear with same sense).
|a+b| >= |a| - |b|
From the same triangle, one side is greater than or equal to the difference of the other two sides. Hence, |a+b| |a| - |b|. Equality holds when a and b are in opposite directions (collinear with opposite sense).
|a-b| <= |a| + |b|
Since a - b = a + (-b), applying the result from (a): |a + (-b)| |a| + |-b| = |a| + |b|. Hence, |a-b| |a| + |b|. Equality holds when a and -b are in the same direction, i.e., a and b are in opposite directions.
|a-b| >= |a|-|b|
Similarly, |a-b| |a| - |-b| = |a| - |b|. Hence, |a-b| |a| - |b|. Equality holds when a and b are in the same direction.
Explanation
The question tests understanding of the triangle law of vector addition and its geometric consequences. The triangle inequality states that in any triangle, the sum of two sides is greater than the third side, and the difference of two sides is less than the third side. When vectors are added geometrically using the triangle law, these properties translate directly into vector magnitude inequalities. The equality conditions depend on whether vectors are collinear and their relative directions.