A standard household vacuum cleaner is unlikely to kill a small spider with big legs. Vacuum cleaners are designed primarily to suck up dust, dirt, and debris from floors and surfaces, and they typically have a collection bag or bin to hold the materials they pick up. When a spider is sucked into the vacuum, it will likely end up in the bag or bin.
Most spiders have a sturdy exoskeleton and can withstand the suction of a vacuum cleaner. Even if a spider is initially harmed by the suction, it may survive inside the bag or bin until it can find a way to escape once the vacuum is turned off.
If you are trying to remove a spider from your home, using a vacuum cleaner is a humane and effective way to do so. However, if you specifically want to kill the spider, it's better to use alternative methods, such as swatting it with a fly swatter or using an insecticide (though this may not be recommended for environmental and health reasons).
Keep in mind that spiders are generally beneficial creatures as they help control insect populations in your home and garden. They are not usually harmful to humans, and most species are not dangerous. If you are concerned about spiders, you can always relocate them outside by capturing them in a container and releasing them outdoors away from your living spaces.