vb.net generate qr code Vectors in Cartesian Three-Space in .NET

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148 Vectors in Cartesian Three-Space
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Dot product of mixed vectors and scalars Suppose that t and u are real numbers, and we have two three-space vectors a and b We can rearrange a dot product of scalar multiples as
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ta ub = tu(a b) The result is always a scalar
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Cross product of mixed vectors and scalars Once again, imagine that t and u are real numbers, and we have two three-space vectors a and b We can rearrange a cross product of scalar multiples as
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ta ub = tu(a b) The result is always a vector quantity
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Imagine two vectors in Cartesian xyz space whose coordinates are expressed as a = (xa,ya,za) and b = (xb,yb,zb) Derive a general expression for a b in the form of an ordered triple
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Let s go back to the concept of SUVs that we learned earlier in this chapter These vectors are i = (1,0,0) j = (0,1,0) k = (0,0,1) Now let s evaluate and list all the cross products we can get from these vectors Using the righthand rule for cross products (from Chap 5) along with the formula for the magnitude of the cross product of vectors, we can deduce, along with the help of Fig 8-7 on page 140, that i j=k j i = k i k = j k i=j j k=i k j = i
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Some More Vector Laws
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We can write the cross product a b as a b = (xa,ya,za) (xb,yb,zb) = (xai + yaj + zak) (xbi + ybj + zbk)
Using the left-hand distributive law for the cross product over vector addition as it applies to trinomials, we can expand this to a b = (xai xbi) + (xai ybj) + (xai zbk) + (yaj xbi) + (yaj ybj) + (yaj zbk) + (zak xbi) + (zak ybj) + (zak zbk) With our newfound knowledge of how scalar multiplication and cross products can be mixed (see Cross product of mixed vectors and scalars ), we can morph each of the terms after the equals sign to get a b = xaxb(i i) + xayb(i j) + xazb(i k) + yaxb(j i) + yayb(j j) + yazb(j k) + zaxb(k i) + zayb(k j) + zazb(k k) A few moments ago, we proved that we always get the zero vector if we take the cross product of any vector with another vector pointing in the same direction That means the cross product of any vector with itself is the zero vector Because the zero vector has zero magnitude, we get the zero vector if we multiply it by any scalar With all this information in mind, we can rewrite the previous equation as a b = 0 + xayb(i j) + xazb(i k) + yaxb(j i) + 0 + yazb(j k) + zaxb(k i) + zayb(k j) + 0 Looking back at the six factoids involving pairwise cross products of i, j, and k, and getting rid of the zero vectors in the previous equation, we can simplify it to a b = xaybk + xazb( j) + yaxb( k) + yazbi + zaxbj + zayb( i) Rearranging the signs, we obtain a b = xaybk xazbj yaxbk + yazbi + zaxbj zaybi This can be morphed a little more, based on rules we ve learned in this chapter, getting a b = (yazb zayb)i + (zaxb xazb)j + (xayb yaxb)k This SUV-based equation tells us three things: The x coordinate of a b is yazb zayb The y coordinate of a b is zaxb xazb The z coordinate of a b is xayb yaxb
150 Vectors in Cartesian Three-Space
Knowing these three facts, we can write the x, y, and z coordinates of a b as an ordered triple to get a b = [(yazb zayb),(zaxb xazb),(xayb yaxb)] We ve found a formula that allows us to directly calculate the cross product of two vectors in xyz space when we re given both vectors as ordered triples
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The formulas for the seven laws in this section were stated straightaway We didn t show how they are derived If you re ambitious (and you have a good pen along with plenty of blank sheets of paper), derive these seven laws by working out the general arithmetic step by step Following are the names of those laws again, for reference: Commutative law for dot product Reverse-directional commutative law for cross product Distributive laws for dot product over vector addition Distributive laws for cross product over vector addition Dot product of cross products Dot product of mixed vectors and scalars Cross product of mixed vectors and scalars
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