104 Vectors and geometry in three dimensions

5 Show that the vectors

U1=cosOsinOi+sinosinOj+cosOk

u2= cosOcos Oi+sin Ocos Oj - sin Ok

u3 = -sin 0 i + cos O j

in the order u1, u2, u3 constitute a right-handed orthonormal system.

6 If

OP1 = 2i + 3j + 2k

OP2=3i+2j+ k

OP3=2i+ j+ k,

find a vector orthogonal to the plane containing Pi, P2, P3. Hint: The vector v

defined by v = P1P2 X P1P3 is an answer, and v = -i + j - 2k.

7 Check the answer to the preceding problem in the following way. Write

the equation of the plane through P1 orthogonal to v and then show that P2 and

P3 lie in this plane.

8 Show that the lines having the equations

x-I y+6 z+10 x-6 y+l z+5

1 - 2 - 3 '^2 -1 = -4

intersect. Then find equations of the line perpendicular to both at their point

of intersection. Solution: The vectors OP and OQ running from the origin to

points P and Q on the two lines are

OP= (1+t)i+(-6+2t)j+(-10+3t)k

OQ = (6 + 2u)i + (-1 - u)j + (-5 - 4u)k,

where t and u are scalars that depend upon P and Q. Equating these vectors

shows that they coincide when t = 3, u = -1 and hence that the given lines

intersect at the point R for which

OR = 4i - k.

Thus R is the point (4,0,-1). This result is easily checked. The vector

i j k

1 2 3

(^2) -1 -4
= -5i + 10j - 5k
is orthogonal to vectors on the given lines and hence the equations
x-4 y-0 z+1
1

-2 - 1
are equations of the required perpendicular line.
9 Prove that each vector v satisfies the equation
i X(v Xi)+j X(v Xj)+k X(v xk) =2v.
10 Show that (I X J) X j 0 1 x (j x j).