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.