10.4 cross product: a vector orthogonal to two given vectors cross product of two vectors in space...
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10.4 Cross product: a vector orthogonal to two given vectors
Cross product of two vectors in space (area of parallelogram)
Triple Scalar product (volume of parallelepiped)
Torque
The Cross Product:Many applications in physics and engineering involve finding a vector in space that is orthogonal to two given vectors. This vector is called the Cross product. (Note that while the dot product was a scalar, the cross product is a vector.)
The cross product of u and v is the vector u x v.
The cross product of two vectors, unlike the dot product, represents a vector.
A convenient way to find u x v is to use a determinant involving vector u and vector v.
1 2 3
1 2 3
i j k
u v u u u
v v v
The cross productis found by taking this determinant.
• The cross product can be expressed as
• Expanding the determinants gives
Vector Products Using Determinants
Now that you can do a cross product the next step is to see why this is useful.
2, 4,5 , 1, 2, 1 6,7, 8u v and u v
Let’s look at the 3 vectors from the last problem
What is the dot product of
2, 4,5 6,7, 8u with u v
And
1, 2, 1 6,7, 8v with u v ?
Recall that whenever two non-zero vectors are perpendicular, their dot product is 0. Thus the cross product creates a vector perpendicular to
the vectors u and v.
?
Vector Products of Unit Vectors
Contrast with scalar products of unit vectors
Signs are interchangeable in cross products
If A & B are vectors, their Vector (Cross) Product is defined as:
A third vector
The magnitude of vector C is AB sinθ where θ is the angle between A & B
• The magnitude of C, which is |A||B| sinθ or |AxB| is equal to the area of the parallelogram formed by A and B.
• The direction of C is perpendicular to the plane formed by A and B
• The best way to determine this direction is to use the right-hand rule
Magnitude of Cross Product
u
v
Area of a parallelogram = bh, in this diagram,
h
sin ,h u
v h
Since sinarea v u
2 vectors in space form a parallelogram
Now you try! Find the area of the triangle with the given verticesA(1, -4, 3) B(2, 0, 2) C(-2, 2, 0)
Calculate the area of the parallelogram PQRS, where P = (1, 1), Q = (2, 3), R = (5, 4), and S = (4, 2)
Geometric application example:
Show that the quadrilateral with vertices at the followingpoints is a parallelogram. Find the area of the parallelogram.Is the parallelogram a rectangle?
A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6)
To begin, plot the vertices, then find the 4 vectors representing the sides of the Parallelogram, and use the property:
Show that the quadrilateral with vertices at the following points is a parallelogram. Find the area of the parallelogram. Is the parallelogram a rectangle? A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6)
x
y
z
Is the parallelogram a rectangle?
Triple Scalar Product or Box Product: For the vectors u, v, and w in space, the dot product of u and is called the triple scalar product of u, v, and w.
A Geometric property of the triple scalar product:
The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given by:
1 2 3
1 2 3
1 2 3
u u u
u v w v v v
w w w
v w
v u v w
A parallelepiped is a figure created when a parallelogram has depth
Example. You Try:
1. Find the volume of a parallelepiped having adjacent edges:
3 5 2 2 3u i j k v j k w i j k
Torque (moment of force)• Tendency of force to rotate an object about an axis, fulcrum, or
pivot.• “twist” of an object• Greek “tau”• Equation where F is force and P is the pivot point.
T=PQ x FWhere magnitude of T measures the tendency of PQ to rotate
counterclockwise about axis directed along T. Another way to look at this is:
Magnitude of torque vector=|r||F|sin x or |r x F| where r is the length of the lever arm and the scalar component of F is
perpendicular to r.
Pg. 748 Torque problemVertical force of 50 pounds applied to a 1-ft lever attached to an axle at P. Find the moment of force about P when θ=60.