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The preceding result might make you wonder, If two vectors point in exactly the same direction or in exactly opposite directions is their cross product always the zero vector The answer is yes, and it doesn t depend on the magnitudes of the original two vectors Let s prove this fact now
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Show that the cross product of any two vectors that point in the same direction or in opposite directions, regardless of their magnitudes, is the zero vector
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When two vectors a and b point in the same direction, the angle qab between them is 0 In such a situation, the magnitude ra b of the cross product is ra b = rarb sin qab = rarb sin 0 = rarb 0 = 0 Therefore, a b = 0, because if a vector has a magnitude of 0, then it s the zero vector by definition When two vectors c and d point in opposite directions, the angle qcd between them is p, so the magnitude rc d of the cross product is rc d = rcrd sin qcd = rcrd sin p = rcrd 0 = 0 Again, we have c d = 0 by definition
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Some More Vector Laws
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Here are some more rules involving vectors You ll find these useful for future reference if you get serious about higher mathematics, physical science, or engineering
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Commutative law for dot product When we figure out the dot product of two vectors, it doesn t matter in which order we work it The result is the same either way If a and b are vectors in three-space, then
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a b=b a
Reverse-directional commutative law for cross product Suppose qab is the angle between two vectors a and b as defined in the plane containing a and b, such that 0 qab p, and such that we re allowed to rotate in either direction The magnitude of the cross-product vector is a nonnegative real number, and is independent of the order in which the operation is performed This can be proven on the basis of the commutative property for multiplication of real numbers We have
ra b = rarb sin qab and rb a = rbra sin qab = rarb sin qab The direction of b a in space is exactly opposite that of a b Figure 8-8 can help us see why this is true when we apply the right-hand rule for cross products (from Chap 5) both ways
Figure 8-8 The vector b a has the same
magnitude as vector a b, but points in the opposite direction
Some More Vector Laws
Distributive laws for dot product over vector addition Imagine that we have three vectors a, b, and c in three-space We can always be sure that
a (b + c) = (a b) + (a c) This fact is called the left-hand distributive law for a dot product over the sum of two vectors It s also true that (a + b) c = (a c) + (b c) which, as you can probably guess, is the right-hand distributive law for the sum of two vectors over a dot product
Distributive laws for cross product over vector addition Suppose that a, b, and c are vectors in three-space Then we can always be sure that
a (b + c) = (a b) + (a c) This property is known as the left-hand distributive law for a cross product over the sum of two vectors A similar rule exists when we cross multiply a sum of vectors on the right The right-hand distributive law for the sum of two vectors over a cross product tells us that (a + b) c = (a c) + (b c) We can expand these rules to pairs of polynomial vector sums, each having n addends (where n = 2, n = 3, n = 4, etc), in the same way as multiplication is distributive with respect to addition for polynomials in algebra For example, for n = 2, we have the cross product of two binomial vector sums, getting (a + b) (c + d) = (a c) + (a d) + (b c) + (b d) In the case of n = 3, the cross product of two trinomial vector sums expands as (a + b + c) (d + e + f ) = (a d) + (a e) + (a f ) + (b d) + (b e) + (b f ) + (c d) + (c e) + (c f )
Dot product of cross products Imagine that we have four vectors a, b, c, and d in three-space We can rearrange a dot product of cross products as
(a b) (c d) = (a c)(b d) (a d)(b c) We always end up with a scalar quantity (that is, a real number)