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Question

# In each case, express the given vector field V in the standard form $\sum v_{i} U_{i}.$ (a) $2 z^{2} U_{1}=7 V+x y U_{3}$ (b) $V(\mathbf{p})=\left(p_{1}, p_{3}-p_{1}, 0\right)_{p}$ for all $\mathbf{p}.$ (c) $V=2\left(x U_{1}+y U_{2}\right)-x\left(U_{1}-y^{2} U_{3}\right)$ (d) At each point $\mathbf{p}, V(\mathbf{p})$ is the vector from the point $\left(p_{1}, p_{2}, p_{3}\right)$ to the point $\left(1+p_{1}, p_{2} p_{3}, p_{2}\right)$ (e) At each point $\mathbf{p}, V(\mathbf{p})$ is the vector from $\mathbf{p}$ to the origin.

Solution

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Step 1
1 of 3

(a)\ Convert the expression in the form $\sum v_i U_i$

\begin{align*} 2z^2 U_1&=7 V+xy U_3\\ 2z^2 U_1-xy U_3&=7V\\ V&=\dfrac{2z^2}{7}U_1-\dfrac{xy}{7}U_3 \end{align*}

(b)\ Convert the given expression in the form $\sum v_i U_i$

\begin{align*} V(p)&=(v_1(p), v_2(p), v_3(p))_p\\ &=p_1 U_1(p)+(p_3-p_1)U_2(p)+(0)U_3(p)\\ &=p_1 U_1(p)+(p_3-p_1)U_2(p) \end{align*}

(c)\ Convert the given expression in the form $\sum v_i U_i$

\begin{align*} V&=2(x U_1+y U_2)-x(U_1-y^2 U_3)\\ &=2x U_1+2 y U_2-xU_1+xy^2 U_3\\ &=x U_1+2 y U_2+xy^2 U_3 \end{align*}

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