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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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142 Vectors in Cartesian Three-Space
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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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An example Let s find the dot product of the two Cartesian vectors
a = (2,3,4) and b = ( 1,5,0) We can call the coordinates xa = 2, ya = 3, za = 4, xb = 1, yb = 5, and zb = 0 Plugging these values into the formula, we get a b = xaxb + yayb + zazb = 2 ( 1) + 3 5 + 4 0 = 2 + 15 + 0 = 13
Another example Suppose we want to find the dot product of the two Cartesian vectors
a = ( 4,1, 3) and b = ( 3,6,6) This time, we have xa = 4, ya = 1, za = 3, xb = 3, yb = 6, and zb = 6 When we substitute these coordinates into the formula, we have a b = xaxb + yayb + zazb = 4 ( 3) + 1 6 + ( 3) 6 = 12 + 6 + ( 18) = 0
Dot Product
Are you confused
Do you wonder how two nonzero vectors can have a dot product of 0 If we look closely at the alternative formula for the dot product, we can figure it out That formula, once again, is a b = rarb cos qab The right-hand side of this equation will attain a value of 0 if at least one of the following is true:
The magnitude of a is equal to 0 The magnitude of b is equal to 0 The cosine of the angle between a and b is equal to 0
Neither of the vectors in the preceding example has a magnitude of 0, so we must conclude that cos qab = 0 That can happen only when a and b are perpendicular to each other, so qab is either p /2 or 3p /2 In the preceding example, the two vectors a = ( 4,1, 3) and b = ( 3,6,6) are mutually perpendicular That s not obvious from the ordered triples, is it
Here s a challenge!
Show that for any two vectors pointing in the same direction, their dot product is equal to the product of their magnitudes Then show that for any two vectors pointing in opposite directions, their dot product is equal to the negative of the product of their magnitudes
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