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vb.net generate qr code < q b qa < p in VS .NET
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(q b, rb) Cross product a b points
3p /2 straight away from us
Figure 55 If qa < qb and the two angles differ by more
than p, then a b points straight away from us as we look down on the plane containing a and b
Cross Product of Two Vectors
If the vectors a and b point in exactly the same direction or in exactly opposite directions, then qb qa = 0 or qb qa = p In these cases, the cross product is the zero vector We ll see why in the next challenge An example Consider the following two polar vectors a and b in standard form: a = (p /4,7) and b = (p,6) Let s find the cross product, a b We have qb qa = p p /4 = 3p /4 Because 0 < qb qa < p, we know that a b points toward us Its magnitude is ra b = rarb sin (qb qa) = 7 6 sin (3p /4) = 7 6 (21/2/2) = 21 21/2 Another example Now let s look at these two polar vectors a and b in standard form and find their cross product a b: a = (p /4,7) and b = (7p /4,6) This time, p < qb qa < 2p, so a b points away from us To calculate the magnitude, we consider the difference angle to be 2p + qa qb = 2p + p /4 7p /4 = p /2 Therefore ra b = rarb sin (2p + qa qb) = 7 6 sin (p /2) = 7 6 1 = 42 86 Vector Multiplication
Are you confused
We haven t discussed how to directly calculate the cross product of two Cartesianplane vectors There s a way to do it, but we must know how to work with vectors in Cartesian threespace We ll learn those techniques in Chap 8 Meanwhile, we can indirectly find the cross product of two Cartesianplane vectors by converting them both to polar form and then finding their cross product the polar way Are you still confused
Here s a game that can help you find the direction of the cross product a b (in that order) between two vectors a and b It involves some maneuvers with your right hand Some mathematicians, engineers, and physicists call this the righthand rule for cross products If 0 < qb qa < p (as in Fig 54), point your right thumb out as if you re making a thumbsup sign Curl your fingers in the counterclockwise rotational sense from a to b Your thumb will point in the general direction of a b If the page on which the vectors are printed is horizontal, your thumb should point straight up If p < qb q a < 2p (as in Fig 55), curl your righthand fingers in the clockwise rotational sense from a to b If the page on which the vectors are printed is horizontal, you ll have to twist your wrist in a clumsy fashion so that your thumb points straight down in the general direction of a b Remember that a b always comes out of the origin precisely perpendicular to the plane containing a and b Here s a challenge! A few moments ago, it was mentioned that if two vectors point in the same direction or in opposite directions, then their cross product is the zero vector Prove it!

