Question

unit vector perpendicular to the resultant of two vector

Answer 1

Compute the cross (vector) product of two vectors
Example: Given a = i − j + 3k and b = −2i + 3j + k, compute a × b.
Solution: a × b is equal to







i j k
1 −1 3
−2 3 1







= (−1 − 9, −(1 + 6), 3 − 2) = (−10, −7, 1).
Find unit vectors perpendicular to two given vectors
Example: Given a = i + 2j + 3k and b = 2i + 3j + 5k, find two unit vectors perpendicular to both
a and b.
Solution: Note that a × b is perpendicular to both a and b. First compute
a × b = (10 − 9, −(5 − 6), 3 − 4) = (1, 1, −1).
Thus the two desired unit vectors are
a × b
|a × b|
=

1
√
3
,
1
√
3
,
−1
√
3

and
−1
√
3
,
−1
√
3
,
1
√
3

.
Compute areas
Example: Find the area of the triangle with vertices P(1, 1, 0), Q(1, 0, 1) and R(0, 1, 1).
Solution: Set a = P Q = (0, −1, 1) and b = P R = (−1, 0, 1). Thus the area is
|a × b|
2
=
|(−1, −1, −1)
|
2 =
√
3
2
.
Compute volumes
Example: Find the volume of the parallelepiped with adjacent edges OP, OQ and OR, where
P(1, 1, 0), Q(1, 0, 1) and R(0, 1, 1) are three points. Also find the volume of the pyramid with
vertices O, P, Q and R.
Solution: Set a = OP = (1, 1, 0), b = OQ = (1, 0, 1) and c = OR = (0, 1, 1). Then the volume of
the parallelepiped is |a · (b × c)|. First compute
a · (b × c) =







1 1 0
1 0 1
0 1 1







= 0 − (0 + 1 + 1) = −2.
Thus the volume of the parallelepiped is 2.
The volume of the pyramid is one sixth of the volume of the parallelepiped, and so it 1
3
.