1. Find the center and radius of the sphere that has (1,−2,4) and (3,4,−12) as endpoints of a diameter.

2. Show that (4, 5, 2), (1, 7, 3), and (2, 4, 5) are vertices of an equilateral triangle.

7. Does the triangle in 3-space with vertices (−1, 2, 3), (2, −2, 0), and (3, 1, −4) have an obtuse angle? Justify your answer.

8. (a). Show that if v = ai+bj is a vector in 2-space, then the vectors v1 = −bi+aj and v2 = bi−aj are both orthogonal to v.

(b). Use the result in part (a) to find two unit vectors that are orthogonal to the vector v = 3i − 2j. Sketch the vectors v, v1, and v2.

9. Find two unit vectors in 2-space that make an angle of 45◦ with 4i + 3j.

10. Show that the direction cosines of a vector satisfy

cos2α + cos2 β + cos2 γ = 1