Question

Question 4.6 Establish the following vector inequalities geometrically or otherwise:

(a) |a + b| ≤ |a| + |b|

(b) |a + b| ≥ ||a| − |b||

(c) |a − b| ≤ |a| + |b|

(d) |a − b| ≥ ||a| − |b||

When does the equality sign above apply?

Chapter Motion In A Plane Page 85

Answer 1

We know ,
|a + b| = √(a + b).(a + b)
= √{a² + b² + 2abcos∅}
= √{|a|² + |b|² +2|a|.|b|cos∅}
For maximum cos∅ = +1
And for minimum cos∅ = -1
e.g
√{|a|² + |b|² 2|a||b|(-1)}≤|a + b| ≤√{|a|² + |b|² + 2|a|.|b|(+1)}
√(|a| - |b|)² ≤ |a + b| ≤√(|a| + |b|)²

(|a| + |b|) > 0 for all real numbers
So, √(|a|+ |b|)² = |a | + |b|

But (|a| - |b|) may be positive or negative .
So, √(|a| - |b|)² = | |a| - |b| |
Hence,

| |a| - |b| | ≤ |a + b| ≤|a| + |b|

(A) |a + b | ≤ |a| + |b | , this equality applies , if a and b are actng in same direction .e.g angle between them = 0 .

(B) |a+ b | ≥ |a| - |b|
See above explanation ,
|a + b | ≥ | |a| - |b| |
When, |a | > |b| then,
|a + b| > |a| - |b| , this equality applies , if a and b are in opposite directions and magnitude of a greater then magnitude of b .

(C) |a - b | = √(|a|² + |b|² + 2|a||b|cos∅)
|a - b| maximum at ∅ = 180°
|a - b| ≤ √{|a| + |b|}² = |a| + |b|
|a - b| ≤ |a| + |b|
Hence, this equality applies , if a and b are in opposite directions .

(D) |a - b| minimum at ∅ = 0°
|a - b| ≥ √{|a|² - |b|}² = | |a| -|b| |
|a - b| ≥| |a| - |b| | , this equality applies , when if a and b are in same direction .

Answer 2

Answer:

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