Simple Nature - Light and Matter

two vectors that we want to multiply can be broken down into components,e.g., (2ˆx + 3ẑ) · ẑ = 2ˆx · ẑ + 3ẑ · ẑ = 0 + 3 = 3. Thusby requiring rotational invariance and consistency with multiplicationof ordinary numbers, we find that there is only one possibleway to define a multiplication operation on two vectors that gives ascalar as the result. 17 The dot product has all of the properties wenormally associate with multiplication, except that there is no “dotdivision.”Dot product in terms of components example 74If we know the components of any two vectors b and c, we canfind their dot product:b · c = ( b x ˆx + b y ŷ + b z ẑ ) · (cx ˆx + c y ŷ + c z ẑ )= b x c x + b y c y + b z c z .Magnitude expressed with a dot product example 75If we take the dot product of any vector b with itself, we findb · b = ( b x ˆx + b y ŷ + b z ẑ ) · (bx ˆx + b y ŷ + b z ẑ )= b 2 x + b 2 y + b 2 z ,so its magnitude can be expressed as|b| = √ b · b .We will often write b 2 to mean b · b, when the context makesit clear what is intended. For example, we could express kineticenergy as (1/2)m|v| 2 , (1/2)mv·v, or (1/2)mv 2 . In the third version,nothing but context tells us that v really stands for the magnitudeof some vector v.Geometric interpretation example 76In figure ae, vectors a, b, and c represent the sides of a triangle,and a = b + c. The law of cosines gives|c| 2 = |a| 2 + |b| 2 − 2|a||b| cos θ .Using the result of example 75, we can also write this asae / The geometric interpretationof the dot product.|c| 2 = c · c= (a − b) · (a − b)= a · a + b · b − 2a · b .Matching up terms in these two expressions, we finda · b = |a||b| cos θ ,17 There is, however, a different operation, discussed in the next chapter, whichmultiplies two vectors to give a vector.Section 3.4 Motion In Three Dimensions 213