Question

Prove that:

| \vec{a} + \vec{b} | \leq | \vec{a} | + | \vec{b} |

State when equality will hold.

Accepted Answer

Case(i): When $\vec{a}, \vec{b}$ are non-collinear vectors;

Let the vectors $\vec{a}, \vec{b}$ be represented by the sides $\vec{O A} and \vec{A B}$ of a triangle $O A B$

Then $\vec{a} + \vec{b} = \vec{O A} + \vec{A B} = \vec{O B}$

Clearly, $O A = |\vec{a}|, A B = |\vec{b}|$ and $O B = |\vec{a} + \vec{b}|$

Since the sum of two sides of a triangle is always greater than the third side.

Therefore, $O A + A B > O B$

$\Rightarrow | \vec{O B} | < | \vec{O A} | + | \vec{A B} |$

$\Rightarrow | \vec{a} + \vec{b} | < | \vec{a} | + | \vec{b} |$

Case(ii): When $\vec{a}, \vec{b}$ are collinear vectors;

Let $\vec{O A} = \vec{a}, \vec{A B} = \vec{b}$

Then $\vec{a} + \vec{b} = \vec{O A} + \vec{A B} = \vec{O B}$ [By triangle law of of vector addition]

Clearly, $O A = |\vec{a}|, A B = |\vec{b}|$ and $O B = |\vec{a} + \vec{b}|$

Now, $O B = O A + A B$

$\Rightarrow |\vec{O B}| = |\vec{O A}| + |\vec{A B}|$

$\Rightarrow | \vec{a} + \vec{b} | = | \vec{a} | + | \vec{b} |$

Hence, in general, $| \vec{a} + \vec{b} | \leq | \vec{a} | + | \vec{b} |$

Hence, proved.

Odisha Board Solutions for Chapter: Vectors, Exercise 1: EXERCISE-12(a)

Odisha Board Mathematics Solutions for Exercise - Odisha Board Solutions for Chapter: Vectors, Exercise 1: EXERCISE-12(a)

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