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(q a, ra)
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Cross product a b points
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3p /2
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straight toward us
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Figure 5-4 If qa < qb and the two angles differ by
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less than p, then a b points straight toward us as we look down on the plane containing a and b
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84 Vector Multiplication
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as ra b), we multiply the original vector magnitudes by each other, and then multiply by the sine of the difference angle Mathematically, ra b = rarb sin (q b qa) In a situation of the sort shown in Fig 5-4, the vector a b points from the coordinate origin straight out of the page toward us If qb qa is larger than p, then things get a little bit complicated To be sure that we assign the correct direction to the vector a b, we must always rotate counterclockwise, and we re never allowed to turn through more than a half circle Figure 5-5 shows an example We rotate through one full circular turn minus qb qa, so the difference angle is 2p (qb qa) which can be more simply written as 2p + qa qb In a situation like this, a b points straight away from us
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p < q b qa < 2 p
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p /2
(q a, ra)
2p + qa q b
(q b, rb) Cross product a b points
3p /2
straight away from us
Figure 5-5 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 Cartesian-plane vectors There s a way to do it, but we must know how to work with vectors in Cartesian three-space We ll learn those techniques in Chap 8 Meanwhile, we can indirectly find the cross product of two Cartesian-plane 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 right-hand rule for cross products If 0 < qb qa < p (as in Fig 5-4), point your right thumb out as if you re making a thumbs-up 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 5-5), curl your right-hand 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!
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