1. Given two vectors such as

A⃗ =2x̂ +4ŷ +7ẑ

B⃗ =−3x̂ +8ŷ −2ẑ

what is their sum? What is its length, and what angle does itmake to the plane of x and y? What is the dot product A⃗⋅B⃗

? What is the dot product of their sum with a unit vector in thez direction (A⃗ +B⃗ )⋅ẑ

? (Hint: this gives the z-component of the sum.).

2. When you apply a force of 100 N to a 25 kg box resting on atable, what is the smallest coefficient of static friction thatwill give a frictional force sufficient to prevent the box frommoving? Suppose you push just a little harder and it starts tomove. If the coefficient of kinetic friction is 0.1, how fast willit be going after 2 seconds of pushing?

3. If you walk up a mountain that is 1.5 km high (1500 meters),and your mass is 65 kg (your weight is about 143 lb ofgravitational force at Earth's surface), how much energy do you usein joules to overcome Earth's gravity? How much would that be infood calories, where 1 calorie is 4.18 J and a \"food\" calorie is akilocalorie in the units used by physicists and chemists.Similarly, if an object of your falls from low Earth orbit toEarth's surface, a distance of say 400 km which is the approximateheight of the International Space Station orbit, how much kineticenergy would it have on reaching the Earth's surface -- assuming(a) it starts from rest, and (b) it does not lose energy to theatmosphere on the way down.