Question

(a) If $\mathbf{u} \cdot \mathbf{r}=\mathbf{v} \cdot \mathbf{r}$ for every vector $\mathbf{r}$ in $V_{\mathbf{r}}$, show that $\mathbf{u}=$ (b) Prove Property 3 of the cross product $$a \times(b+c)=a \times b+a \times c$$ by showing that $$[a \times(b+c)] \cdot r=[a \times b+a \times c] \cdot r$$ for every vector $\mathbf{r}$ in $V$,

Name: Problem 41, Chapter 9: Calculus
Uploaded: 2021-09-05T19:22:24-08:00
Duration: 2 min 4 s
Channel: Carson Merrill
Description: (a) If 𝐮 ·𝐫=𝐯 ·𝐫 for every vector 𝐫 in V𝐫, show that 𝐮= (b) Prove Property 3 of the cross product a ×(b+c)=a ×b+a ×c by showing that [a ×(b+c)] ·r=[a ×b+a ×c] ·r for every vector 𝐫 in V,

   (a) If $\mathbf{u} \cdot \mathbf{r}=\mathbf{v} \cdot \mathbf{r}$ for every vector $\mathbf{r}$ in $V_{\mathbf{r}}$, show that $\mathbf{u}=$
(b) Prove Property 3 of the cross product
$$a \times(b+c)=a \times b+a \times c$$
by showing that
$$[a \times(b+c)] \cdot r=[a \times b+a \times c] \cdot r$$
for every vector $\mathbf{r}$ in $V$,

Calculus

James Stewart 4th Edition

Chapter 9, Problem 41

Instant Answer

Solved by Expert Carson Merrill

09/05/2021

Step 1

Step 1: Given that $\mathbf{u} \cdot \mathbf{r}=\mathbf{v} \cdot \mathbf{r}$ for every vector $\mathbf{r}$ in $V_{\mathbf{r}}$, we can subtract $\mathbf{v} \cdot \mathbf{r}$ from both sides to get $\mathbf{u} \cdot \mathbf{r} - \mathbf{v} \cdot \mathbf{r} = 0$. Show more…

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(a) If $\mathbf{u} \cdot \mathbf{r}=\mathbf{v} \cdot \mathbf{r}$ for every vector $\mathbf{r}$ in $V_{\mathbf{r}}$, show that $\mathbf{u}=$ (b) Prove Property 3 of the cross product $$a \times(b+c)=a \times b+a \times c$$ by showing that $$[a \times(b+c)] \cdot r=[a \times b+a \times c] \cdot r$$ for every vector $\mathbf{r}$ in $V$,

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Key Concepts

Nondegeneracy of the Dot Product

The dot product in Euclidean vector spaces is nondegenerate, meaning that if the dot product of a vector with every other vector is zero, then that vector must be the zero vector. This property is crucial for establishing vector equality: if (u - v)·r = 0 for every vector r, then u - v must be the zero vector, implying u = v.

Bilinearity and Distributivity of the Cross Product

The cross product is a bilinear operation, which means it is linear in each of its arguments. In particular, it is distributive over addition. This property can be demonstrated by showing that the dot products of a × (b + c) and a × b + a × c with any arbitrary vector are identical. This approach exploits the fundamental relationship between the cross product and the scalar triple product to validate the distributive law.

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