# asp.net barcode reader free What are the three standard unit vectors (SUVs) in Cartesian xyz space in Visual Studio .NET Creation Code 3/9 in Visual Studio .NET What are the three standard unit vectors (SUVs) in Cartesian xyz space

What are the three standard unit vectors (SUVs) in Cartesian xyz space
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The three SUVs in Cartesian xyz space are defined as the standard-form vectors i = (1,0,0) j = (0,1,0) k = (0,0,1) Any Cartesian xyz space vector in standard form can be split up into a sum of scalar multiples of the three SUVs The scalar multiples are the coordinates of the ordered triple representing the vector For example, suppose we have a = (xa,ya,za)
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We can break the vector a up in the following manner: a = (xa,ya,za) = (xa,0,0) + (0,ya,0) + (0,0,za) = xa(1,0,0) + ya(0,1,0) + za(0,0,1) = xai + yaj + zak
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Suppose we have two standard-form vectors in Cartesian xyz space, defined as a = (xa,ya,za) and b = (xb,yb,zb) How can we calculate the dot product a b How can we calculate the dot product b a How do they compare
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We can calculate a b as a real number using the formula a b = xaxb + yayb + zazb Alternatively, it is a b = rarb cos qab where ra is the magnitude of a, rb is the magnitude of b, and qab is the angle between the vectors as determined in the plane containing them both, rotating from a to b In the same fashion, we can calculate b a using the formula b a = xbxa + ybya + zbza Alternatively, it is b a = rbra cos qba where rb is the magnitude of b, ra is the magnitude of a, and qba is the angle between the vectors as determined in the plane containing them both, rotating from b to a The dot product is commutative In other words, for all vectors a and b in Cartesian xyz space, we can be sure that
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a b=b a
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Review Questions and Answers Question 8-9
How can we find the cross product of two standard-form vectors a and b in three-space if we know their magnitudes and the angle between them
The cross product a b is a vector perpendicular to the plane containing both a and b, and whose magnitude ra b is given by ra b = rarb sin qab where ra is the magnitude of a, rb is the magnitude of b, and qab is the angle between a and b, expressed in the rotational sense going from a to b We should define the angle so that it s always within the range 0 qab p If we look at a and b from some point far outside of the plane containing them, and if qab turns through a half circle or less counterclockwise as we go from a to b, then the crossproduct vector a b points toward us If qab turns through a half circle or less clockwise as we go from a to b, then a b points away from us
Question 8-10
Imagine that we have two vectors in xyz space whose coordinates are a = (xa,ya,za) and b = (xb,yb,zb) How can we express a b as an ordered triple