If a^{\to} and b^{\to} are two vectors such that |a^{\to}+b^{\to}| = |a^{\to}|, then prove that vector 2a^{\to}+b^{\to} is perpendicular to vector b^{\to}

CBSE Class 12 Math 2013 Solved Paper - Question 20

CBSE Class 12 Math 2013 Solved Paper

Question : 20 of 29

Marks: +1, -0

a↖{→}

and

b↖{→}

are two vectors such that

|a↖{→}+b↖{→}|

|a↖{→}|

, then prove that vector

2a↖{→}+b↖{→}

is perpendicular to vector

b↖{→}

Solution:

|a↖{→}+b↖{→}|

|a↖{→}|

⇒

|a↖{→}+b↖{→}|^2

|a↖{→}|^2

⇒

|a↖{→}|^2

2a↖{→}.b↖{→}

|b↖{→}|^2

|a↖{→}|^2

⇒

2a↖{→}.b↖{→}

|b↖{→}|^2

= 0 ... (1)
Now,

2a↖{→}.b↖{→}

b↖{→}

2a↖{→}.b↖{→}

b↖{→}.b↖{→}

2a↖{→}.b↖{→}

|b↖{→}|^2

= 0
We know that if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.
Thus,

2a↖{→}+b↖{→}

is perpendicular to

b↖{→}

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