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Assertion (A) : For any two vectors vec(...

Assertion (A) : For any two vectors `vec(a)` and `vec(b)`, we always have `|vec(a)+vec(b)|le|vec(a)|+|vec(b)|`
Reason (R ) : The given inequality holds trivially when either `vec(a)=0` or `vec(b)=0` i.e., in such a case
`|vec(a)+vec(b)|=0=|vec(a)|+|vec(b)|`.
Then consider

A

Both A and R are true and R is NOT the correct explanation of A

B

Both A and R are true but R is NOT the correct explanation of A

C

A is true but R is false

D

A is false but R is True

Text Solution

AI Generated Solution

The correct Answer is:
To solve the assertion and reason problem, we need to analyze the statement given in the assertion (A) and the reason (R) step by step. ### Step 1: Understanding the Assertion The assertion states that for any two vectors \(\vec{a}\) and \(\vec{b}\), the following inequality holds: \[ |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \] This is known as the triangle inequality for vectors. ### Step 2: Proving the Assertion To prove the assertion, we can use the properties of vectors and the definition of magnitude. 1. **Triangle Inequality**: By the triangle inequality, for any two vectors \(\vec{a}\) and \(\vec{b}\), the inequality \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) holds true. This can be shown geometrically or algebraically. 2. **Geometric Interpretation**: If you visualize \(\vec{a}\) and \(\vec{b}\) as two sides of a triangle, then the length of the third side (which is \(|\vec{a} + \vec{b}|\)) cannot exceed the sum of the lengths of the other two sides (which are \(|\vec{a}|\) and \(|\vec{b}|\)). Thus, the assertion is true. ### Step 3: Understanding the Reason The reason states that the inequality holds trivially when either \(\vec{a} = 0\) or \(\vec{b} = 0\). 1. **Case when \(\vec{a} = 0\)**: If \(\vec{a} = 0\), then: \[ |\vec{a} + \vec{b}| = |\vec{b}| \quad \text{and} \quad |\vec{a}| + |\vec{b}| = 0 + |\vec{b}| = |\vec{b}| \] Thus, \( |\vec{a} + \vec{b}| = |\vec{b}| \) which satisfies the inequality. 2. **Case when \(\vec{b} = 0\)**: Similarly, if \(\vec{b} = 0\), then: \[ |\vec{a} + \vec{b}| = |\vec{a}| \quad \text{and} \quad |\vec{a}| + |\vec{b}| = |\vec{a}| + 0 = |\vec{a}| \] Thus, \( |\vec{a} + \vec{b}| = |\vec{a}| \) which also satisfies the inequality. Therefore, the reason is also true, but it does not provide a complete explanation for the assertion since the triangle inequality holds for all vectors, not just when one of them is zero. ### Step 4: Conclusion Both the assertion (A) and the reason (R) are true. However, the reason does not adequately explain the assertion, as the triangle inequality is a general property that applies to all vectors, not just in the trivial cases. ### Final Answer - Assertion (A) is true. - Reason (R) is true but does not explain (A).
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