"As an example, canon 6d full frame has the highest resolution of 5472 x 3648 with a pixel aspect ratio of 3:2 . As this is the highest I assume this is utilising the full size of the sensor."

Yes, and no. The actual sensor is slightly larger than the effective highest resolution. The sensor is 21 MP while the used area is 20 MB (source). That means that almost the full size is used.

There are a few pixels at the edges that are not used. There can be several reasons for that, for example that the sensor doesn't have the exact 3:2 aspect ratio, or that the extra pixels are used to adjust the used area to better fit what you see in the viewfinder.

"If the canon 6d is recording video the highest resolution is 1920 x 1080, with a diffrent aspect ratio of 16:9. Does this mean it is cropping the full frame sensor(in this case cropping the top and bottom for a more rectangular sized sensor field of view)?"

Yes.

Although there might be a few more pixels available on the sides of the 3:2 area, they are far from enough to reach the 16:9 ratio. It's likely that the camera doesn't use the pixels outside the 3:2 area and just crops the image for other aspect ratios.

"if this is the case, then will using a low resolution with the same pixel aspect ratio (3:2) as the highest resolution be a way to utilise the full size (field of view) of the sensor?"

Yes.