11.4 The Cross product
For an animation of this topic visit:
/
http://www.math.umn.edu/~nykamp/m2374/readings/crossprod
Red x Green = Blue
What is the cross product?
• a × b is a vector that is perpendicular to both a
and b.
• ||a × b|| is the area of the parallelogram
spanned by a and b (i.e. the parallelogram
whose adjacent sides are the vectors a and b).
• The direction of a×b is determined by the righthand rule. (This means that if we curl the
fingers of the right hand from a to b, then the
thumb points in the direction of a × b.)
Definition of Cross Product of
Two Vectors in Space
Find the cross product of vectors (by hand and with a
calculator)
u = 3i +2j – k
v = i + 2k
Note: To find a cross product on the TI89
Press 2nd 5 (math) – 4 matrix – L Vector ops - crossP
crossP([3,2,-1],[1,0,2])
Definition of Cross Product of
Two Vectors in Space
For an explanation and animation of the cross
product visit:
http://www.math.umn.edu/~nykamp/m2374/readings/
crossprod/
Example 1
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•
•
•
•
Find the cross product of the two vectors
u = i – 2j + k v = 3i + j – 2k
Find u x v
Find v x u
Find v x v
Example 1
•
•
•
•
•
Find the cross product of the two vectors
u = i – 2j + k v = 3i + j – 2k
Find u x v
Find v x u
Find v x v
Example 1 continued
• u = i – 2j + k v = 3i + j – 2k
• Find a unit vector that is orthogonal to u
and v
The Triple Scalar Product
Geometric Property of Triple Scalar
Example 5
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u = 3i – 5j + k
v = 2j – 2k
w = 3i + j + k
Find the volume of
the parallelepiped
having vectors u, v
and w for sides
Example 5 hint
This works because bxc yields a vector perpendicular to a and b
in with the magnitude of the area of the parallelogram formed (the
base of the parallel piped)
bxc points in the direction of the height of the parallelepiped when
dotted with a this gives the magnitude of bxc times the portion of
a that points in the direction of bxc (the direction of the height )in
other words this gives us the same as the formula Bh
Q: Why should you never make a math teacher angry?
A: You might get a cross product
Q: What do you get when you cross an elephant
and a banana?
A: | elephant | * | banana | * sin(theta)
Proof of the cross product
proof that the cross product is orthogonal to the two original
vectors is part of the homework
│u x v │ = │u │ │v │sinθ