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Question :

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||

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Last updated date: 20th Jun 2024
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Answer
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Hint: You can establish these vector inequalities by using the properties of vector addition and the triangle inequality.


Step-by-Step Solutions:

(a) |a+b| < |a| + |b|:

Explanation: This inequality is known as the Triangle Inequality.

Proof: Geometrically, if you have vectors 'a' and 'b,' and you form the vector 'a + b' by connecting the tail of 'b' to the head of 'a,' then the magnitude of 'a + b' represents the length of the diagonal of the parallelogram formed by 'a' and 'b.' The sum of the magnitudes of 'a' and 'b' represents the lengths of two sides of the parallelogram. By the triangle inequality, the length of the diagonal (|a + b|) is always less than the sum of the lengths of the two sides (|a| + |b|).


(b) |a+b| > ||a| − |b||:

Explanation: This inequality also involves the Triangle Inequality.

Proof: To establish this inequality, consider the vectors 'a' and '-b' (which is the negation of vector 'b'). Now, when you add 'a' and '-b,' you effectively subtract 'b' from 'a.' The magnitude of 'a - b' represents the length of the diagonal of the parallelogram formed by 'a' and '-b.' The magnitude of ||a| - |b|| represents the difference between the magnitudes of 'a' and 'b.' By the triangle inequality, the length of the diagonal (|a - b|) is always greater than the absolute difference between the magnitudes (||a| - |b||).


(c) |a−b| < |a| + |b|:

Explanation: This is another application of the Triangle Inequality.

Proof: Geometrically, if you have vectors 'a' and '-b' (or 'a' and 'b' if you consider their directions), the magnitude of 'a - b' represents the length of the diagonal of the parallelogram formed by 'a' and '-b.' The sum of the magnitudes of 'a' and 'b' represents the lengths of two sides of the parallelogram. By the triangle inequality, the length of the diagonal (|a - b|) is always less than the sum of the lengths of the two sides (|a| + |b|).


(d) |a−b| > ||a| − |b||:

Explanation: This inequality also involves the Triangle Inequality.

Proof: To establish this inequality, consider the vectors 'a' and 'b.' The magnitude of 'a - b' represents the length of the diagonal of the parallelogram formed by 'a' and 'b.' The magnitude of ||a| - |b|| represents the absolute difference between the magnitudes of 'a' and 'b.' By the triangle inequality, the length of the diagonal (|a - b|) is always greater than the absolute difference between the magnitudes (||a| - |b||).


Note: The Triangle Inequality is a fundamental concept in vector mathematics and geometry. It's used to prove various inequalities involving vector magnitudes and is a key tool in analyzing vectors and their properties.

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