Question

True or False? If a*b=a*c, then either a=0 or b=c. The answer is false but I want to know how to get there with work. These are vectors, it has arrows above each letter and 0.

Answer 1

assuming these are dot products, if a dot b = a dot c, then both dot products could be equal to zero

ie

a dot b = 0

a dot c = 0

which would mean that a dot b = a dot c is true

Answer 2

this would mean that if 'a' is a vector that's perpendicular to both vectors b and c, then

a dot b = a dot c

is true

but b = c doesn't necessarily have to be true and 'a' could be a nonzero vector

Answer 3

a*b=a*c
a*b-a*c=0
a*(b-c)=0
either a=0 orb-c=0,
which gives a=0, or b=c
a is parallel b-c
or vector a is parallel to the plane containing the vectors b and c

Answer 4

here*=cross product

Answer 5

From a=0 or b=c, what's the next step to show that the statement is false?

Answer 6

just pick any two vectors b, c where they aren't equal to each other

then find the cross product of those vectors b and c to get the vector a

You'll most likely find that 'a' is a nonzero vector

Answer 7

So a doesn't =0 and b doesn't =c, right?

Answer 8

yeah both are possible and you can still have a*b=a*c true (* is the dot product)

Answer 9

Thanks.