Question 1160832: Let L be the line in R^3 that consists of all scalar multiples of the vector w = (2, 2, 1). Find the reflection of the vector v=(4, 7, 1) in the line L.
Answer by CPhill(2303)   (Show Source):
You can put this solution on YOUR website! ```python
import numpy as np
w = np.array([2, 2, 1], dtype=float)
v = np.array([4, 7, 1], dtype=float)
# Projection of v onto w:
# proj_L(v) = (v . w / w . w) * w
v_dot_w = np.dot(v, w)
w_dot_w = np.dot(w, w)
proj = (v_dot_w / w_dot_w) * w
# Reflection across a line L:
# refl_L(v) = 2 * proj_L(v) - v
refl = 2 * proj - v
print(f"v . w = {v_dot_w}")
print(f"w . w = {w_dot_w}")
print(f"proj = {proj}")
print(f"refl = {refl}")
```
```text
v . w = 23.0
w . w = 9.0
proj = [5.11111111 5.11111111 2.55555556]
refl = [6.22222222 3.22222222 4.11111111]
```
```python
refl_fraction = (56/9, 29/9, 37/9)
print("56/9:", 56/9)
print("29/9:", 29/9)
print("37/9:", 37/9)
print("Norm squared v:", 4**2 + 7**2 + 1**2)
print("Norm squared refl:", (56**2 + 29**2 + 37**2)/81)
```
```text
56/9: 6.222222222222222
29/9: 3.2222222222222223
37/9: 4.111111111111111
Norm squared v: 66
Norm squared refl: 66.0
```
To find the reflection of the vector $\mathbf{v}$ in the line $L$, we use the formula:
$$\operatorname{refl}_L(\mathbf{v}) = 2\,\operatorname{proj}_L(\mathbf{v}) - \mathbf{v}$$
---
### Step 1: Compute the Orthogonal Projection of $\mathbf{v}$ onto $L$
The projection of $\mathbf{v} = (4, 7, 1)$ onto the direction vector $\mathbf{w} = (2, 2, 1)$ is:
$$\operatorname{proj}_L(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{w}}{\Vert{}\mathbf{w}\Vert{}^2} \mathbf{w}$$
1. Compute the dot product $\mathbf{v} \cdot \mathbf{w}$:
$$\mathbf{v} \cdot \mathbf{w} = (4)(2) + (7)(2) + (1)(1) = 8 + 14 + 1 = 23$$
2. Compute $\Vert{}\mathbf{w}\Vert{}^2$:
$$\Vert{}\mathbf{w}\Vert{}^2 = 2^2 + 2^2 + 1^2 = 4 + 4 + 1 = 9$$
3. Calculate the projection:
$$\operatorname{proj}_L(\mathbf{v}) = \frac{23}{9} (2, 2, 1) = \left( \frac{46}{9}, \frac{46}{9}, \frac{23}{9} \right)$$
---
### Step 2: Compute the Reflection
Substitute $\operatorname{proj}_L(\mathbf{v})$ and $\mathbf{v}$ into the reflection formula:
$$\operatorname{refl}_L(\mathbf{v}) = 2 \left( \frac{46}{9}, \frac{46}{9}, \frac{23}{9} \right) - (4, 7, 1)$$
$$\operatorname{refl}_L(\mathbf{v}) = \left( \frac{92}{9}, \frac{92}{9}, \frac{46}{9} \right) - \left( \frac{36}{9}, \frac{63}{9}, \frac{9}{9} \right)$$
Subtracting each component:
* **$x$-component:** $\frac{92 - 36}{9} = \frac{56}{9}$
* **$y$-component:** $\frac{92 - 63}{9} = \frac{29}{9}$
* **$z$-component:** $\frac{46 - 9}{9} = \frac{37}{9}$
---
### Result
$$\operatorname{refl}_L(\mathbf{v}) = \left( \frac{56}{9}, \frac{29}{9}, \frac{37}{9} \right)$$
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