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Precalculus with Limits: A Graphing Approach, Sixth Edition
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Precalculus with Limits: A Graphing Approach, Sixth Edition View details
4. Vectors and Dot Products
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Exercise 1 Page 440

Recall the definition of the dot product.

Yes

We are asked if the following equation is true for any vectors u and v. u* v? =v* u

Let's first calculate the dot products for a pair of example vectors u=⟨ 1, 3⟩ and v=⟨ 2, 4⟩ .

u* v v* u
⟨ 1, 3⟩*⟨ 2, 4⟩= 1( 2)+ 3( 4) =14 ⟨ 2, 4⟩*⟨ 1, 3⟩= 2( 1)+ 4( 3) =14
Here, we can see that u* v is equal to u* v. This is also true for any other pair of vectors u and v. We call this the Commutative Property of the Dot Product. u* v=v* u âś“

Extra

 Formal Proof
Consider the vectors u=⟨ u_1,u_2⟩ and v=⟨ v_1,v_2⟩. We will try to show that u* v is equal to u* v.
u* v

a * b = a_1 * b_1 + a_2 * b_2

u_1v_1+u_2v_2
CommutativePropMult

Commutative Property of Multiplication

v_1u_1+v_2y_2

a * b = a_1 * b_1 + a_2 * b_2

v* u
Since the vectors u and v are arbitrary, the given equation is always true. u* v=v* u âś“
Subchapter links expand_more
arrow_right Vocabulary and Concept Check p.440
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arrow_right Procedures and Problem Solving p.440-442
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arrow_right Conclusions p.442
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arrow_right Cumulative Mixed Review p.442
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