the cross product third type of multiplying vectors
TRANSCRIPT
The Cross ProductThe Cross ProductThe Cross ProductThe Cross Product
Third Type of Multiplying VectorsThird Type of Multiplying Vectors
Cross Products
1 1 1 2 2 2
1 2 2 1 1 2 2 1 1 2 2 1
If v and
are two vectors in space, the cross product
v is defined as the vector
v ( ) ( ) ( )
a i b j c k w a i b j c k
w
w b c b c i a c a c j a b a b k
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Determinants• It is much easier to do this using
determinants because we do not have to memorize a formula.
• Determinants were used last year when doing matrices
• Remember that you multiply each number across and subtract their
products
Finding Cross Products Using Equation
2 3
3 2 Find v
( 3 1 2 1 (2 ( 1) 3 1) (2 ( 2) ( 3 3)
(3 2) ( 2 3) ( 4 9)
5 5 5
v i j k
w i j k w
i j k
i j k
i j k
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Evaluating a Determinant
2 32 2 1 3 4 3 1
1 2
6 5(6 1) 5 2 6 10 4
2 1
Evaluating Determinants
1 4 2 4 2 12 1 4
3 1 1 1 1 31 3 1
(1 1 3 4) 2 1 1 4 2 3 1 1
11 2 5
A B C
A B C
A B C
A B C
Using Determinants to Find Cross Products
• This concept can help us find cross products.
• Ignore the numbers included in the column under the vector that will be inserted when setting up the determinant.
Using Determinants to Find Cross Products
• Find v x w given• v = i + j• w = 2i + j + k
Using Determinants to Find Cross Products
1 0 1 0 1 11 1 0
1 1 2 1 2 12 1 1
(1 0) (1 0) (1 2)
i j k
i j k
i j k
i j k
Using Determinants to Find Cross Products
• If v = 2i + 3j + 5k and w = i + 2j + 3k,
• find • (a) v x w• (b) w x v• (c) v x v
Using Determinants to Find Cross Products
( )
2 3 5
1 2 3
3 5 2 5 2 3
2 3 1 3 1 2
(9 10) (6 5) (4 3)
a
i j k
v w
i j k
i j k
i j k
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Using Determinants to Find Cross Products
( )
1 2 3
2 3 5
2 3 1 3 1 2
3 5 2 5 2 3
(10 9) (5 6) 3 4
Notice that this is the same values as but
the signs are opposite. (Neg. is now pos. and vice
versa)
b
i j k
w v
i j k
i j k
i j k
v w
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Using Determinants to Find Cross Products
2 3 5
2 3 5
3 5 2 5 2 3
3 5 2 5 2 3
(15 15) (10 10) (6 6)
0
This leads us to one of the properties of cross
products.
i j k
v v
i j k
i j k
Algebraic Properties of the Cross Product
• If u, v, and w are vectors in space and if is a scalar, then
• u x u = 0• u x v = -(v x u)• (u x v) = (u) x v = u x (v)• u x (v + w) = (u x v) + (u x w)
Examples• Given u = 2i – 3j + k v = -3i + 3j
+ 2k• w = i + j + 3k• Find• (a) (3u) x v• (b) v . (u x w)
Examples(3 ) 3(2 3 ) 6 9 3
(3 ) 6 9 3
3 3 2
9 3 6 3 6 9
3 2 3 2 3 3
( 18 9) (12 ( 9) (18 27)
27 21 9
u i j k i j k
i j k
u v
i j k
i j k
i j k
Examples
2 3 1
1 1 3
3 1 2 1 2 3
1 3 1 3 1 1
( 9 1) (6 1) (2 ( 3))
10 5 5
( 3 ( 10)) (3 ( 5)) (2 5)
30 15 10 25
i j k
u w
i j k
i j k
i j k
v u w
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Geometric Properties of the Cross Product
• Let u and v be vectors in space• u x v is orthogonal to both u and v.• ||u x v|| = ||u|| ||v|| sin where is
the angle between u and v.• ||u x v|| is the area of the
parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides
Geometric Properties of the Cross Product
• u x v = 0 if and only if u and v are parallel.
Finding a Vector Orthogonal to Two Given
Vectors• Find a vector that is orthogonal to • u = 2i – 3j + k and v = i + j + 3k
• According to the preceding slide, u x v is orthogonal to both u and v. So to find the vector just do u x v
Finding a Vector Orthogonal to Two Given
Vectors
2 3 1
1 1 3
3 1 2 1 2 3
1 3 1 3 1 1
( 9 1) (6 1) (2 ( 3))
10 5 5
i j k
u v
i j k
i j k
i j k
Finding a Vector Orthogonal to Two Given
Vectors • To check to see if the answer is
correct, do a dot product with one of the given vectors. Remember, if the dot product = 0 the vectors are orthogonal
Finding a Vector Orthogonal to Two Given
Vectors
(2 10) ( 3 5) (1 5)
0
u u v
Finding the Area of a Parallelogram
• Find the area of the parallelogram whose vertices are P1 = (0, 0, 0),
• P2 = (3,-2, 1), P3 = (-1, 3, -1) and • P4 = (2, 1, 0)
• Two adjacent sides of this parallelogram are u = P1P2 and v = P1P3.
Finding the Area of the Parallelogram
1 2
1 3
1 2 1 3
1 2 1 3
2 2 2
3 0, 2 0,1 0 3 2
( 1 0,3 0, 1 0) 3
3 2 1 2 7
1 3 1
Area of a Parallelogram =
1 2 7 1 4 49 54 3 6
PP i j k
PP i j k
i j k
PP PP i j k
PP PP
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Your Turn• Try to do page 653 problems 1 –
47 odd.