Question

If two vectors a and b are such that |vectors a|=2, |vectors b|=1 and vectors a . b=1, then find the value of vectors (3a-5b).(2a+7b).

Answer 1

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We have,

a+b=ca+b=c ……………………(1)

Also,

|a|+|b|=|c|…………(2)

Square the equation, (1), we get,

(a+b).(a+b)=c.c

Implies,

|a|2+|b|2|+2(a.b)=|c| ……….(3)

From (2), we can get,

|a|2+|b|2|+2|a||b|=|c|2 ………(4)

Since the RHS’s of (3) and (4) are same, we can equate them, we get,

|a|2+|b|2|+2(a.b)=|a|2+|b|2|+2|a||b|

Implies,

a.b=|a||b|

Dot Product of 2 vectors is the product of absolute values of the vectors with the cosine of angle between them. So,

|a||b|cosx=|a||b|

Here, xx is the angle between aa and bb.

So,

cosx=1

Take cosine inverse or arccosarccos on both sides, we get,

x=cos−1(1)

Implies,

x=0


So, the angle between the 2 given vectors is 0, which means they’re parallel.

I hope my answer was helpful.
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