(We call this the scalar product because the product is a scalar rather than a vector.) Given two vectors A = a 1 x + a 2 y and B = b 1 x + b 2 y, the scalar product A ? B is the following: But what if we want to calculate the component of some vector in the direction of another arbitrary vector? To this end, we define the scalar product (also called the dot product) of two vectors. We can factor out the magnitudes of A and B ( A and B, respectively) and write the scalar product in terms of these magnitudes and the scalar product of two corresponding unit vectors, a and b, which are in the directions of A and B, respectively. Now, let's consider these two unit vectors by way of the diagram below. Note that we have defined the angle between the vectors as ?. Note that the unit vector a is composed of a component vector in the direction of b and a component vector perpendicular to the direction of b. Using right triangle trigonometry, we see that the component of a in the direction of b is cos ? (the magnitude, or "length," of a is unity). We need to now prove that this result is the same as the scalar product of a and b. We'll draw two arbitrary unit vectors as shown below. Notice that we have defined two angles: the angle ? between a and b, and ? between a and the x-axis. Let's write both a and b in terms of their component parts in the x and y directions (corresponding to the x and y unit vectors, respectively). We'll calculate the scalar product of these two vectors as follows. Let's use the properties of trigonometric functions (specifically, addition formulas) to simplify the result. Practice Problem: A projectile of mass 5 kilograms has a velocity 10 x + 15 y meters per second. What is the component of the projectile's momentum in the direction of 4 x + 3 y? Solution: First, let's calculate the momentum p of the projectile. Now calculate the scalar product of p and b: To find the component of the momentum in the direction of the vector 4 x + 3 y, we need to calculate the scalar product of p and the unit vector corresponding to 4 x + 3 y (we'll call this vector B). Thus, the component of the momentum in the direction of 4 x + 3 y is 85 newton seconds (the corresponding vector would be 85 b newton seconds). This is compared with the magnitude of p, which is about 90.1 newton seconds. Practice Problem: Show that the scalar product of two perpendicular vectors is always zero. Solution: Consider any two vectors A and B where the angle between them is 90° (thus making them perpendicular). The formula for the scalar product of these vectors is the following, which was derived earlier. Let's insert the angle 90° into the formula. Thus, by virtue of the fact that cos 90° is zero, the scalar product of any two perpendicular vectors is zero. We typically think of work as something that involves an expenditure of effort-for instance, lifting a heavy box might be referred to as hard work. The concept of work in physics is similar work in this context is defined as the product of the force applied to an object and the distance that the object is displaced (moved). And if you’re looking for a jacket that’s slightly heavier and a bit less breathable, Outdoor Research’s Uberlayer is also on sale and is even cheaper ( $173) than the Ascendant.For example, to lift the heavy box, one must apply an upward force to it over a certain distance if the box needs to be lifted only a little bit, less work is done than if the box needs to be lifted high. If you’ve been eyeing up one of Outdoor Research’s award-winning jackets with active insulation, now’s your chance as the Ascendant is currently 25 percent off at Backcountry. Aside from the benefits of keeping one in your kit year-round, summer is also the best time to score deals on some of the year’s finest outerwear. You might be asking why we’d even bother talking about an insulated jacket in the dead of summer. It’s incredibly packable and fits well underneath a shell while you take those deep blower turns you hiked up for in the first place. The jacket insulates your body only where it needs it - so, less sweaty back - while also shedding wind. The outer shell is constructed from breathable and stretchy Pertex that is also highly water resistant. The insulation sits directly on whatever you’re wearing underneath the jacket. It utilizes Polartec Alpha Direct insulation, which has no liner. The Ascendants is a game changer when it comes to active insulation. Even in the cold, ascents are fated to be sweaty endeavors, especially when you’re wearing a loaded pack. Outdoor Research’s Ascendant jacket lessens the price you pay on the way up. You still have to get to the top before that reward can be reaped. And like most endurance activities, it is, to a certain extent - but in touring, the pain only increases the feeling of reward when your bindings are locked and the only direction left to go is down. Unless you’ve done it, ski touring could seem like a practice in misery.
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