For any two vectors \overrightarrow{a} and ...

CBSE Class 12 Math 2024 All Sets Solved Paper

Question : 8 of 20

Marks: +1, -0

For any two vectors

{a}↖{→}

and

{b}↖{→}

, which of the following statements is always true ?

Solution:

Let us assume

{a}↖{→} \ne \vec 0

and

{b}↖{→} \ne \vec 0

{a}↖{→}. {b}↖{→} = | {a}↖{→} | {b}↖{→} | \cos θ

Taking modulus on both sides then we get,

| {a}↖{→}. {b}↖{→} | = | {a}↖{→} | {b}↖{→} | \cos θ |

| \cos θ | = \;{ | {a}↖{→}.{b}↖{→} | }/{ | {a}↖{→}||{b}↖{→} | }

As we know that

0 ≤ | \cos θ | ≤ 1

\;{ | {a}↖{→}.{b}↖{→} | }/{ | {a}↖{→}||{b}↖{→} | }≤ 1

Therefore,

| {a}↖{→} .{b}↖{→} | ≤ | {a}↖{→} | {b}↖{→} |

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