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
Answer 8-9
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
Answer 8-10
We can plug in the coordinate values directly into the formula a b = [( yazb zayb),(zaxb xazb),(xayb yaxb)]
9
Question 9-1
How do we determine the cylindrical coordinates of a point in three-space
Answer 9-1
We paste a polar plane onto a Cartesian xy plane, creating a reference plane The positive Cartesian x axis is the reference axis To determine the cylindrical coordinates of a point P, we first locate its projection point, P on the reference plane:
Part One
The direction angle q is expressed counterclockwise from the reference axis to the ray that goes out from the origin through P The radius r is the distance from the origin to P The height h is the vertical displacement (positive, negative, or zero) from P to P The basic scheme is shown in Fig 10-7 We express the cylindrical coordinates of our point of interest as an ordered triple: P = (q,r,h)
Question 9-2
Can we have nonstandard direction angles in cylindrical coordinates Can we have negative radii Are there any restrictions on the values of the height coordinate
Answer 9-2
Theoretically, we can have a nonstandard direction angle But if we come across that situation, it s best to add or subtract whatever multiple of 2p will bring the direction angle into the preferred range 0 q < 2p If q 2p, it represents at least one complete counterclockwise rotation from the reference axis If q < 0, it represents clockwise rotation from the reference axis We can have a negative radius in theoretical terms However, if we come across that sort of situation, it s best to reverse the direction angle and then consider the radius positive If r < 0, we can take the absolute value of the negative radius and use it as the radius coordinate Then we must add or subtract p to or from q to reverse the direction, while also making sure that the new angle is larger than 0 but less than 2p
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