In this sentence, why need to use "towards" and not "toward"? As "responsibilities" is a plural noun already.
A girl with big dreams but with even bigger responsibilities towards her family.
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The choice between toward and towards is generally a matter of style and region. Toward is more common in US English, while towards is more common in UK English.
to·ward | \ ˈtō-ərd, ˈtȯ(-ə)rd\
variants: or towards \ ˈtō-ərd(z), ˈtȯ(-ə)rd(z), tə-ˈwȯrd(z), ˈtwȯrd(z) \
Note that use of variants. That means that the variant is accepted, but it's not as common. This is also stated explicitly under its adjective definition:
1 or less commonly towards
Meanwhile, the definition of towards used by Lexico (Oxford) shows:
(North American toward)
And under its definition of toward:
variant of towards
So while there is no absolute rule about this (it still comes down to style), it's generally more common to use toward in the US and towards in the UK.
Most dictionary examples will tell you that you should follow the word with either "to" or "for", depending on the direction of the responsibility.
"Responsibility towards...." is used a whole lot less. This ngram shows that it is barely used in comparison to the examples above.
Examples using "towards" tend to be moral responsibilities towards concepts, such as a responsibility towards society, or a responsibility towards the environment. I would say that you should only really use this expression when referring to similar concepts - where we are not under strict compulsion to carry out a responsibility, but where our thoughts and actions should perhaps be oriented to that cause.