1. Analytic Geometry in Three Dimensions10
Analytic Geometry in Three Dimensions
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2. The Cross Product of Two Vectors
10.3
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3. What You Should Learn Find cross products of vectors in space
Use geometric properties of cross products of vectors in space
Use triple scalar products to find volumes of parallelepipeds

4. The Cross Product

5. The Cross Product
Many applications in physics, engineering, and geometry involve finding a vector in space that is orthogonal to two given vectors.
In this section, you will study a product that will yield such a vector. It is called the cross product, and it is most conveniently defined and calculated using the standard unit vector form.

6. Example 1 – Finding Cross Products
Given u = i + 2j + k and v = 3i + j + 2k, find each cross product.
u  v
v  u
v  v
Solution:

7. Example 1 – Solution = (4 – 1)i – (2 – 3)j + (1 – 6)k = 3i + j – 5k
cont’d
= (4 – 1)i – (2 – 3)j + (1 – 6)k
= 3i + j – 5k
= (1 – 4)i – (3 – 2)j + (6 – 1)k

8. Example 1 – Solution = –3i – j + 5k
cont’d
= –3i – j + 5k
Note that this result is the negative of that in part (a).

9. The Cross Product

10. Geometric Properties of the Cross Product

11. Geometric Properties of the Cross Product
The first Algebraic Properties of the Cross Product indicates that the cross product is not commutative. In particular, this property indicates that the vectors u  v and v  u have equal lengths but opposite directions. The following list gives some other geometric properties of the cross product of two vectors.

12. Example 2 – Using the Cross Product
Find a unit vector that is orthogonal to both
u = 3i – 4j + k and v = –3i + 6j.
Solution:
The cross product u  v, as shown in Figure 10.18, is orthogonal to both u and v.
= –6i – 3j + 6k
Figure 10.18

13. Example 2 – Solution Because = 9
cont’d
Because
= 9
a unit vector orthogonal to both u and v is

14. The Triple Scalar Product

15. The Triple Scalar Product
For the vectors u, v and w in space, the dot product of u and v  w is called the triple scalar product of u, v and w.

16. The Triple Scalar Product
When the vectors u, v and w do not lie in the same plane, the triple scalar product
u  (v  w)
can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure

17. Example 4 – Volume by the Triple Scalar Product
Find the volume of the parallelepiped having
u = 3i – 5j + k, v = 2j – 2k, and w = 3i + j + k
as adjacent edges, as shown in Figure
Solution:
The value of the triple scalar product is
Figure 10.21

18. Example 4 – Solution = 3(4) + 5(6) + 1(–6) = 36.
cont’d
= 3(4) + 5(6) + 1(–6)
= 36.
So, the volume of the parallelepiped is
| u  (v  w) | = | 36 |
= 36.