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11.3: Torque and Cross-Product
Definitions: The torque of a vector F about a vector r
The cross-product of vectors a and b:
Note that a × b is orthogonal to both a and b: If a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i, and a × b =
hx1 , x2 , x3 i, then we have:
The cross-product of vectors a and b (ONLY in R3 ) is given by
a×b=
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NOTES:
1. Simple calculation method:
2. Geometric significance:
3. |a × b| =
4. Useful Properties (all listed on p668):
Examples:
Find a × b if a = i − j + 2k and b = 2i − 3j − 2k
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Given the points P (1, 1, −1), Q(2, 0, −1), and R(1, −1, 1) find a vector orthogonal to the plane containing these points.
Find the area of 4P QR.
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Find the volume of the parallelipiped whose corner is formed by the vectors a = h2, −1, 4i, b =
h1, −3, 0i, and c = h3, 1, −2i.
On Beyond Average:
If a, b, and c are 3-dimensional vectors, determine if each of the following is a vector, a scalar, or
meaningless:
(a × b) · c
(a · b) × c
(a · b)c
(a × b) × c
a · (b × c)
The vectors a, b, and c = b − a all lie in the same plane as shown in the diagram below:
Circle all TRUE statements below:
a×b=0
(a × b) · c = 0
a × b points into the page
b × (a × c) points in the direction of −a
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