In Vectors, If a.b = a.c, is it necessary that b = c. Please explain?

Answer 1

Let x and y be two vectors
x.y is the scalar product which is defined as-

x.y=|x| |y| sin Z

| | is the magnitude

Z is the angle between the two.
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a.b=a.c

=>|a| |b| sin m =|a| |c| sin n

m :angle between a and b vectors
n :angle between a and c vectors

So we get,by dividing both sides by |a|,

|b| sin m =|c| sin n

Unless m=n,u cannot say that |b|=|c|

If m=n, |b|=|c|

Also ,b and c make same angle(m=n) with vector a
So,
vector b =vector c


In general, a.b=a.c does not mean b=c
as we cannot directly cancel a without taking the scalar product of which a is a part

Answer 2

Consider the vectors 5 and 1 inclined at angle 15 degree and vectors 5 and 2 at angle 61.1 degree.

a.b = ab cos15 = 4.83

a.c = ac cos 61.1 =4.83

But b is not equal to c.

Answer 3

b and c has to be the same in order for this equation to be true.

a.b = a.c

example:

a = 4
b = 5
c = 5

a.5 = a.5
4.5 = 4.5
4+4+4+4+4 = 4+4+4+4+4
20 = 20

Answer 4

split the vector into the two components

x: a = a
y: b = c

don't know what exactly you need to know!
but there is no explaining necessary

Answer 5

You have a equality. You can simplify dividing the two sides by same factor.
if you divide by a , you obtain b=c

Answer 6

If a=0
b=c is not necessary