Given two vectors $u = i + 3j + k$ and $v = 4i + 7j$, the problem is to find the cross product $u \times v$.

1. Problem Description

Given two vectors $u = i + 3j + k$ and $v = 4i + 7j$, the problem is to find the cross product $u \times v$.

2. Solution Steps

The cross product of two vectors $u = u_1i + u_2j + u_3k$ and $v = v_1i + v_2j + v_3k$ is given by:

$u \times v = (u_2v_3 - u_3v_2)i - (u_1v_3 - u_3v_1)j + (u_1v_2 - u_2v_1)k$

Here, $u = i + 3j + k$, which means $u_1 = 1$, $u_2 = 3$, and $u_3 = 1$.

And $v = 4i + 7j$, which means $v_1 = 4$, $v_2 = 7$, and $v_3 = 0$.

Using the formula for the cross product:

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

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

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

3. Final Answer

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