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and b = (xb,yb,zb) = xbi + ybj + zbk When we add these components straightaway, we get a + b = xai + yaj + zak + xbi + ybj + zbk The commutative law for vector addition allows us to rearrange the addends on the righthand side of this equation to get a + b = xai + xbi + yaj + ybj + zak + zbk Now let s use the right-hand distributive law for multiplication of the sum of two scalars by a vector to morph the previous equation into a + b = (xa + xb)i + (ya + yb)j + (za + zb)k That s the sum of the original vectors, expressed as a sum of multiples of SUVs
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A specific example Suppose we re given a vector b = ( 2,3, 7), and we re told to break it into a sum of multiples of i, j, and k We can imagine i as going 1 unit to the right, j as going 1 unit upward, and k as going 1 unit toward us The breakdown proceeds as follows:
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b = ( 2,3, 7) = 2 (1,0,0) + 3 (0,1,0) + ( 7) (0,0,1) = 2i + 3j + ( 7)k = 2i + 3j 7k
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By now you might wonder, Must I memorize all of the rules mentioned in this section Not necessarily You can always come back to these pages for reference But honestly, I recommend that you do memorize them If you take a lot of physics or engineering courses later on, you ll be glad that you did
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As we ve been doing throughout this chapter, let s revisit our generic standard-form vectors in xyz space, defined as a = (xa,ya,za)
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and b = (xb,yb,zb) We can calculate the dot product a b as a real number using the formula a b = xaxb + yayb + zazb Alternatively, it is a b = rarb cos qab where ra is the magnitude of a, rb is the magnitude of b, and qab is the angle between a and b as determined in the plane containing them both, rotating from a to b
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