Does the hand used for magic matter?

Question

“You do,” said Dumbledore. “So you will need to hold on to my arm very tightly. My left, if you don’t mind — as you have noticed, my wand arm is a little fragile at the moment.”

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"Well, now — Mr. Potter. Let me see.” He pulled a long tape measure with silver markings out of his pocket. “Which is your wand arm?”

“Er — well, I’m right-handed,” said Harry.

There definitely is such a thing as a "wand hand/arm" but does it really make a difference which hand you use? And is the wand arm always the same as the dominant one? Are there specific left-handed wands? (see Are there left-handed wizards in Harry Potter? for more on this subject.)

Answer 1

I do not think that there is an explicit answer to this question anywhere in canon, at least that I know of. The only thing that it can be compared to is the dominant hand (as Himarm mentioned in the comments) in our muggle world. If you are a right handed person, you can learn, rather easily, how to throw a ball consistently with your right hand. However, if you were to try to throw the ball with your left hand, the mechanics are likely to be off (comparable to wand movements [swish and flick]) and the resulting throw is likely to be sloppy, not and off target.

Being off target, along with improper wand motions, is likely going to hinder the caster of a spell as they are not as likely to hit their target, or to create the proper wand motion necessary for the spell. This significantly affects the ability of the caster to be effective with their spell work, but does not necessarily affect their ability to cast a spell.

All of that said, I cannot imagine why a witch/wizard would not be able to, with practice, learn how to use their non-dominant hand with enough proficiency to be able to comfortably, and accurately cast using either hand. This would be especially true of those witches or wizards who may have lost their dominant arm to dark magic, as Moody did with his leg.

Answer 2

I believe Dumbledore's reference to his weak arm was due to the fact it had become withered and all but unusable after he sacrificed it to destroy the horcrux in Voldemort's ring.

As for the power of magic itself, it's pretty evident throughout HP that both the way you say the words (It's Leviooooosa) AND the way you move your wand (Swish and Flick) when casting a spell play a big part in how successful, stable and powerful your spell is. If you cast a spell with your non-dominant hand, it stands to reason that your technique would not be as refined, and thus your spell would not be quite as potent.

And I'm pretty sure that there ARE both Left-handed and Ambidextrous wizards in the HP universe, just as there are Lefties and Ambies in amongst muggles.