1. 设 P 和 Q 是球体上截然相反的点，而 R 是球体上的任何其他点。展示
PR 与 QR 成直角。

2. 证明顶点 ~r1、~r2 和 ~r3 的三角形面积等于
12 j(~r2 ~r1) (~r3 ~r1)j。

3. 求平面 Ax + By + Cz + D = 0 之间的角度（的余弦或正弦）和

4. 求（余弦或正弦）平面之间的角度 Ax + By + Cz + D = 0 和
ax+by+cz +d = 0。这些平面什么时候垂直？它们什么时候平行？在这

5. 两条平行线由方程~r = ~r1 + t~a 和~r = ~r2 + t~a 给出。找出

6. 使用 j~a ~ bj = ab sin 和 ~a ~ b = ab cos 的事实来证明

7. (a) 在分量中使用直接计算，表明
(~a ~ b) ~c = (~a ~c)~ b (~ b ~c)~a:

(b) 从 (a) 部分和三重积的适当性质推导出
(~a ~ b) (~c ~ d) = (~a ~c)(~ b ~ d) (~a ~ d)(~ b ~c)：

8. 证明以顶点为原点的圆锥，单位向量~k为其对称轴，

(ii) 对于复数 a 和 b，证明并几何解释这个结果。

Suggested Homework Problems

1. Let P and Q be diametrically opposite points and R any other point on a sphere. Show
that PR meets QR at right angles.
HINT: Choose the unit sphere, and the north and south pole for P and Q, respectively.

2. Show that the area of the triangle with vertices ~r1, ~r2, and ~r3 equals
12 j(~r2 ~r1)  (~r3 ~r1)j.

3. Find the (cosine or sine of) the angle  between the plane Ax + By + Cz + D = 0 and
the line x = x0 + t, y = y0 + t, and z = z0 + t.

4. Find (cosine or sine of) the angle  between the planes Ax + By + Cz + D = 0 and
ax+by+cz +d = 0. When are these planes perpendicular? When are they parallel? In this
case, what is the distance between them?

5. Two parallel lines are given by the equations ~r = ~r1 + t~a and ~r = ~r2 + t~a. Find the
distance between them.

6. Use the facts that j~a ~ bj = ab sin  and ~a ~ b = ab cos  to show that

7. (a) Using a direct calculation in components, show that
(~a ~ b) ~c = (~a  ~c)~ b (~ b  ~c)~a:

(b) Deduce from part (a) and the appropriate property of the triple product that
(~a ~ b)  (~c  ~ d) = (~a  ~c)(~ b  ~ d) (~a  ~ d)(~ b  ~c):

8. Show that the cone with the vertex at the origin, the unit vector ~ k as its symmetry axis,
and the aperture angle of =4 is described by the equation x2

(ii) For complex a and b, show that and interpret this result geometrically.

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