Given two vectors $u = i + 3j + k$ and $v = 4i + 7j$, the problem is not clearly stated. I will assume the question asks to find the cross product of the two vectors. We are to find $u \times v$.

1. Problem Description

Given two vectors $u = i + 3j + k$ and $v = 4i + 7j$, the problem is not clearly stated. I will assume the question asks to find the cross product of the two vectors. We are to find $u \times v$.

2. Solution Steps

First, we write the vectors $u$ and $v$ in component form:

$u = <1, 3, 1>$

$v = <4, 7, 0>$

The cross product $u \times v$ is given by the determinant:

$u \times v = \begin{vmatrix} i & j & k \\ 1 & 3 & 1 \\ 4 & 7 & 0 \end{vmatrix}$

Expanding the determinant:

$u \times v = i \begin{vmatrix} 3 & 1 \\ 7 & 0 \end{vmatrix} - j \begin{vmatrix} 1 & 1 \\ 4 & 0 \end{vmatrix} + k \begin{vmatrix} 1 & 3 \\ 4 & 7 \end{vmatrix}$

$u \times v = i(3(0) - 1(7)) - j(1(0) - 1(4)) + k(1(7) - 3(4))$

$u \times v = i(0 - 7) - j(0 - 4) + k(7 - 12)$

$u \times v = -7i + 4j - 5k$

3. Final Answer

$u \times v = -7i + 4j - 5k$

or $<-7, 4, -5>$