To find the new coordinates of the point (0, 4, 5) after shifting the origin to (1, 2, -3), we can follow these steps:
### Step 1: Identify the original point and the new origin
- The original point is given as \( P(0, 4, 5) \).
- The new origin after shifting is \( O'(1, 2, -3) \).
### Step 2: Use the formula for new coordinates
To find the new coordinates \( (x', y', z') \) of the point with respect to the new origin, we can use the following formulas:
- \( x' = x - x_1 \)
- \( y' = y - y_1 \)
- \( z' = z - z_1 \)
Where \( (x, y, z) \) are the coordinates of the original point and \( (x_1, y_1, z_1) \) are the coordinates of the new origin.
### Step 3: Substitute the values into the formulas
Here, we have:
- \( x = 0 \), \( y = 4 \), \( z = 5 \)
- \( x_1 = 1 \), \( y_1 = 2 \), \( z_1 = -3 \)
Now, we can substitute these values into the formulas:
1. For \( x' \):
\[
x' = 0 - 1 = -1
\]
2. For \( y' \):
\[
y' = 4 - 2 = 2
\]
3. For \( z' \):
\[
z' = 5 - (-3) = 5 + 3 = 8
\]
### Step 4: Write the new coordinates
After calculating the new coordinates, we find:
- The new coordinates of the point \( P(0, 4, 5) \) with respect to the new origin \( O'(1, 2, -3) \) are \( P'(-1, 2, 8) \).
### Final Answer
The new coordinates of the point (0, 4, 5) with respect to the new frame are \( (-1, 2, 8) \).
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