answer:You know, as an ice climber, I'm always on the hunt for the next great adventure. I've been fortunate enough to have climbed in some of the most incredible locations around the world, but there are a few places that really stand out to me. One of my all-time favorites is the Ruth Glacier in Alaska's Denali National Park. There's something truly awe-inspiring about climbing in the shadow of North America's highest peak. The Ruth Glacier is this stunning, sprawling expanse of ice that stretches as far as the eye can see, with towering seracs and crevasses that seem to shift and change with every step. And the ice itself is just so pure and blue – it's like nothing I've ever seen anywhere else. Another spot that really gets my blood pumping is the Baltoro Glacier in the Karakoram range of Pakistan. This place is just a climber's paradise – massive peaks, endless ice fields, and some of the most challenging and technical climbing routes on the planet. And the scenery? Just breathtaking. You're surrounded by these towering giants of mountains, with glaciers cascading down their flanks like frozen waterfalls. It's truly humbling to be in the presence of such natural beauty and power. And then there's Patagonia – specifically, the glaciers of El Chaltén and Los Glaciares National Park. This place is like the wild west of ice climbing – untamed, unpredictable, and full of surprises. The glaciers here are so dynamic, with massive calving events and crevasses that seem to open up overnight. It's a real test of your skills and your nerves, but the rewards are well worth it. There's nothing quite like the thrill of standing on the ice, feeling the wind whipping through your hair, and knowing that you're one of the few people on earth who's ever set foot in this incredible place. For me, it's not just about the climbing itself – it's about immersing myself in these incredible environments, and feeling the power and majesty of the natural world. There's something deeply spiritual about being on the ice, surrounded by the silence and the beauty of these frozen landscapes. It's a feeling that's hard to put into words, but it's what keeps me coming back, time and time again.

answer:Crevasses – they're one of the biggest hazards we face as ice climbers and glacier travelers. Essentially, a crevasse is a deep, narrow opening or fracture in the surface of a glacier. They're formed when the glacier ice is subjected to stress and tension, causing it to crack and split open. Crevasses can be incredibly deep – I've seen some that drop down 100 feet or more into the darkness. And they can be hidden, too – sometimes they're covered by a thin layer of snow or ice, making them almost invisible. That's when they're most hazardous, because you can be walking along, thinking everything is fine, and then suddenly you're plunging down into this abyss. The problem with crevasses is that they're not just static holes in the ice – they're dynamic, moving features that can change shape and size rapidly. As the glacier flows, the crevasses can open and close, making it difficult to predict where they'll be and how deep they'll be. And when you fall into one, it's like being swallowed by the glacier itself – the walls are sheer, the ice is hard as rock, and it's almost impossible to climb out without help. That's why crevasse rescue is such a critical skill for glacier travelers. When someone falls into a crevasse, every minute counts – the longer you're down there, the greater the risk of hypothermia, injury, or even death. As a rescuer, you have to move fast, using specialized equipment and techniques to stabilize the victim, anchor yourself to the surface, and then pull them up to safety. It's a high-pressure situation that requires a lot of training, experience, and focus. But even with the risks, I love the challenge of navigating crevasses. It's like solving a puzzle, trying to read the glacier's terrain and anticipate where the crevasses will be. And when you're able to successfully rescue someone from a crevasse, it's an incredible feeling – you've just given them a second chance at life, and that's a pretty powerful thing.

answer:My crevasse rescue kit is something I take very seriously – it's a lifeline when things go wrong on the glacier. Here are some of the essential items I always carry with me: First and foremost, I've got a good rope – a 60-meter dynamic rope that's specifically designed for crevasse rescue. This rope is the backbone of any rescue operation, and I make sure it's always in good condition and easily accessible. Next, I've got a set of pulleys and carabiners – these are the workhorses of crevasse rescue. The pulleys allow me to change the direction of the rope and multiply the force I apply, while the carabiners provide a secure attachment point to the rope and the anchor. Speaking of anchors, I always carry a set of ice anchors – V-threads, ice screws, and deadman anchors. These allow me to secure the rope to the surface of the glacier and prevent the victim from falling further. I also carry a set of prusiks – short lengths of cord that can be attached to the rope to create a makeshift handhold or foothold. These are crucial for hauling the victim up to the surface. In addition to the technical gear, I've got a first aid kit specifically designed for crevasse rescue. This includes items like tourniquets, hemostatic agents, and warm blankets to help stabilize the victim and treat any injuries. Communication is also key in a crevasse rescue, so I always carry a two-way radio and a whistle to signal for help. And, of course, I've got a headlamp and extra batteries – crevasse rescues can take place at any time of day or night, and good lighting is essential. Finally, I've got a small repair kit with duct tape, cordage, and other essentials to fix any gear that gets damaged during the rescue. The last thing you want is to have a critical piece of gear fail in the middle of a rescue. Overall, my crevasse rescue kit is designed to be lightweight, portable, and comprehensive – it's a carefully curated collection of gear that's designed to help me respond to any situation that might arise on the glacier.

answer:Given mathbf{u} times mathbf{v} + 2mathbf{u} = mathbf{w} and mathbf{w} times mathbf{u} = 2mathbf{v}, [(mathbf{u} times mathbf{v} + 2mathbf{u}) times mathbf{u} = 2mathbf{v}.] Expanding, we get: [(mathbf{u} times mathbf{v}) times mathbf{u} + 2mathbf{u} times mathbf{u} = 2mathbf{v}.] Since mathbf{u} times mathbf{u} = mathbf{0}, we have: [(mathbf{u} times mathbf{v}) times mathbf{u} = 2mathbf{v}.] Using the vector triple product: [mathbf{u} times (mathbf{v} times mathbf{u}) = (mathbf{u} cdot mathbf{u}) mathbf{v} - (mathbf{u} cdot mathbf{v}) mathbf{u}.] Since mathbf{u} is a unit vector, we have: [mathbf{v} - (mathbf{u} cdot mathbf{v}) mathbf{u} = 2mathbf{v}.] From this, it must be that mathbf{u} cdot mathbf{v} = 0. Now, we compute mathbf{u} cdot (mathbf{v} times mathbf{w}) as follows: begin{align*} mathbf{u} cdot (mathbf{v} times mathbf{w}) &= mathbf{u} cdot (mathbf{v} times (mathbf{u} times mathbf{v} + 2mathbf{u})) &= mathbf{u} cdot (mathbf{v} times (mathbf{u} times mathbf{v})) + 2mathbf{u} cdot (mathbf{v} times mathbf{u}). end{align*} Since mathbf{v} times (mathbf{u} times mathbf{v}) = (mathbf{v} cdot mathbf{v}) mathbf{u} - (mathbf{v} cdot mathbf{u}) mathbf{v} = mathbf{u} and mathbf{u} cdot (mathbf{v} times mathbf{u}) = 0 (as mathbf{v} times mathbf{u} is orthogonal to mathbf{u}), [mathbf{u} cdot (mathbf{v} times mathbf{w}) = mathbf{u} cdot mathbf{u} + 2 times 0 = 1.] Thus, mathbf{u} cdot (mathbf{v} times mathbf{w}) = boxed{1}.