138 Vectors in Cartesian Three-Space

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Here are some fundamental laws that apply to vectors and real-number scalars in xyz space We won t delve into the proofs Most of these facts are intuitive, and resemble similar laws in algebra We ve already seen a couple of them, but they re repeated here so you can use this section for reference in the future Keep in mind that all of these rules assume that the vectors are in standard form

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Commutative law for vector addition When we add any two vectors in xyz space, it doesn t matter in which order the addition is done The resultant vector is the same either way If a and b are vectors, then

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a+b=b+a

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Commutative law for vector-scalar multiplication When we find the product of a vector and a scalar in xyz space, it doesn t matter which way we do it If a is a vector and k is a scalar, then

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ka = ak

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Associative law for vector addition When we add up three vectors in xyz space, it makes no difference how we group them If a, b, and c are vectors, then

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(a + b) + c = a + (b + c)

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Associative law for vector-scalar multiplication Suppose that we have two scalars k1 and k2, along with some vector a in Cartesian xyz space If we want to find the product k1k2a, it makes no difference how we group the quantities We can write this rule mathematically as

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k1k2a = (k1k2)a = k1(k2a)

Distributive laws for scalar addition Imagine that we have some vector a in xyz space, along with two real-number scalars k1 and k2 We can always be sure that

a(k1 + k2) = ak1 + ak2 and (k1 + k2)a = k1a + k2a The first rule is called the left-hand distributive law for multiplication of a vector by the sum of two scalars The second law is called the right-hand distributive law for multiplication of the sum of two scalars by a vector

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Distributive laws for vector addition Suppose we have two vectors a and b in xyz space, along with a real-number scalar k We can always be certain that

k(a + b) = ka + kb and (a + b)k = ak + bk The first rule is called the left-hand distributive law for multiplication of a scalar by the sum of two vectors The second law is called the right-hand distributive law for multiplication of the sum of two vectors by a scalar

Unit vectors Let s take a close look at the structures of two different vectors a and b in xyz space, both of which are expressed in the standard form Suppose that their coordinates can be written as the familiar generic ordered triples

a = (xa,ya,za) and b = (xb,yb,zb) Either of these vectors can be split up into a sum of three component vectors, each of which lies along one of the coordinate axes The component vectors are scalar multiples of mutually perpendicular vectors with magnitude 1 We have 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) and b = (xb,yb,zb) = (xb,0,0) + (0,yb,0) + (0,0,zb) = xb(1,0,0) + yb(0,1,0) + zb(0,0,1) The three vectors (1,0,0), (0,1,0), and (0,0,1) are called standard unit vectors (We can call them SUVs for short) It s customary to name them i, j, and k, such that i = (1,0,0) j = (0,1,0) k = (0,0,1)

140 Vectors in Cartesian Three-Space

Figure 8-7 The three standard unit vectors i, j, and k in Cartesian xyz space

Figure 8-7 illustrates the coordinates and direction angles of the three SUVs in Cartesian three-space, where each axis division represents 1/5 of a unit Note that each SUV is perpendicular to the other two

A generic example Let s see what happens when we add two generic vectors component-by-component Again, suppose we have

a = (xa,ya,za) and b = (xb,yb,zb) Expressed as sums of multiples of the SUVs, these two vectors are a = (xa,ya,za) = xai + yaj + zak