Yes – but you have to be careful. You would have to put the video production business through a separate legal entity – e.g. a separate Limited Company.
As a Sole Trader you cannot just decide to charge VAT on one income stream but not on another – as HMRC would say there is only one legal entity (you) and it is VAT registered. So, if you undertake both businesses as a sole Trader then you should charge VAT on all sales.
It may be that you are not sure how successful the video production will be, so you don’t necessarily want to set it up as a separate Limited Company right away. In that case, you could put the video production business through your sole-trader business and “swallow the VAT” yourself rather than passing it on to your video customers. If you decide later that it is a viable business, you could put the business through a Limited Company (you might even consider doing that with your printing business, too).