The problem provides two vectors, $u$ and $v$, defined as $u = -i + 3j + k$ and $v = 4i + 7j$. This implies $u = <-1, 3, 1>$ and $v = <4, 7, 0>$.

1. Problem Description

The problem provides two vectors, $u$ and $v$, defined as $u = -i + 3j + k$ and $v = 4i + 7j$. This implies $u = <-1, 3, 1>$ and $v = <4, 7, 0>$.

2. Solution Steps

Since the problem doesn't specify what to compute, let us compute the dot product of u and v ($u \cdot v$).

The dot product of two vectors $u = <u_1, u_2, u_3>$ and $v = <v_1, v_2, v_3>$ is given by:

$u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$

In our case, $u = <-1, 3, 1>$ and $v = <4, 7, 0>$. Thus,

$u \cdot v = (-1)(4) + (3)(7) + (1)(0)$

$u \cdot v = -4 + 21 + 0$

$u \cdot v = 17$

3. Final Answer

The dot product $u \cdot v = 17$.