Find the vector, not with determinants, but by using properties of cross products.

k × (i − 8j)

using the associative property, this is just

k×i - k×8j
Now recall that
k×i = j
k×j = -i
...

To find the cross product of k and (i - 8j) without using determinants, you can use the properties of cross products.

The cross product of two vectors, u = u1i + u2j + u3k and v = v1i + v2j + v3k, can be calculated using the following formula:

u × v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k

Let's apply this formula to the given vectors k and (i - 8j):

k × (i - 8j) = (0)(-8) - (0)(1)i - (1)(-8)j + (1)(1)k
= 0i + 8j + k
= 8j + k

Therefore, the cross product of k and (i - 8j) is 8j + k.

To find the cross product of two vectors using properties of cross products, you can expand the expression using the distributive property and then apply the cross product properties.

Given: k × (i - 8j)

We can expand the expression as follows:

k × (i - 8j) = k × i - k × 8j

The cross product between any unit vector and k is zero, since they are parallel. Therefore:

k × i = 0

Now we're left with:

0 - k × 8j

To find the cross product of k and 8j, we can use the cross product properties. The cross product of any two vectors, a × b, can be calculated by taking the determinant of a matrix, where the rows correspond to the components of a and b.

In this case, k and 8j can be written as:

k = (0, 0, 1)
8j = (0, 8, 0)

Now we can set up the matrix for the determinant:

| i j k |
| 0 8 0 |
| 0 0 1 |

To calculate the determinant, we expand along the first row:

| 8 0 |
| 0 1 |

The determinant is given by:

det = (8 * 1) - (0 * 0) = 8

So the cross product of k and 8j is 8i.

Therefore, the vector k × (i - 8j) is equal to 8i.