Page 179 - Advanced engineering mathematics

6.3 The Cross Product 159

To obtain a vector in the direction of u and of length d,divide u by its length to obtain a unit
vector, then multiply this vector by d. Therefore,

u u · v
proj v = d = u.
u
u u 2
As an example, suppose v = 4i − j + 2k and u = i − j + 2k. Then
2
u · v = 9 and u = 6,
so
9 3
proj v = u = (i − j + 2k).
u
6 2
If we think of these vectors as forces, we may interpret proj v as the effect of v in the
u
direction of u.

SECTION 6.2 PROBLEMS

In each of Problems 1 through 6, compute the dot product 8. (−1,0,0),i − 2j
of the vectors and the cosine of the angle between them.
Also determine if the vectors are orthogonal. 9. (2,−3,4),8i − 6j + 4k
10. (−1,−1,−5),−3i + 2j
1. i,2i − 3j + k
2. 2i − 6j + k,i − j 11. (0,−1,4),7i + 6j − 5k
3. −4i − 2i + 3k,6i − 2j − k 12. (−2,1,−1),4i + 3j + k
4. 8i − 3j + 2k,−8i − 3j + k
5. i − 3k,2j + 6k In each of Problems 13, 14, and 15, find the projection of
v onto u.
6. i + j + 2k,i − j + 2k
In each of Problems 7 through 12, find the equation of
13. v = i − j + 4k,u =−3i + 2j − k
the plane containing the given point and orthogonal to the
given vector. 14. v = 5i + 2j − 3k,u = i − 5j + 2k
7. (−1,1,2),3i − j + 4k 15. v =−i + 3j + 6k,u = 2i + 7j − 3k

6.3 The Cross Product

The dot product produces a scalar from two vectors. The cross product produces a vector
from two vectors.
Let F = a 1 i + b 1 j + c 1 k and G = a 2 i + b 2 j + c 2 k.The cross product of F with G is the
vector F × G defined by
F × G = (b 1 c 2 − b 2 c 1 )i + (a 2 c 1 − a 1 c 2 )j + (a 1 b 2 − a 2 b 1 )k.

Here is a simple device for remembering and computing these components. Form the determinant

i j k

a 1 b 1 c 1

a 2 b 2 c 2
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October 14, 2010 14:21 THM/NEIL Page-159 27410_06_ch06_p145-186