Complete the square to write the following equation in the standard form for a sphere. ​ ​

Given and , find the vector component of u orthogonal to v. ​

Find the vector v whose initial and terminal points are given below. (6,2), (2,–2)  

Given , find .  

Find the distance between the planes given below.  

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.  

Given and , find . ​

Find the unit vector in the direction of u.  

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place.  

Find the magnitude of the vector given below.