Step 1: Define the Vectors
Let the vectors be:
- Vector a = \( \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} \)
- Vector b = \( -3\mathbf{i} + 6\mathbf{j} + 2\mathbf{k} \)
Step 2: Calculate the Magnitudes
To find the angle bisector, first compute the magnitudes of the vectors:
- Magnitude of a:
\[ \| \mathbf{a} \| = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
- Magnitude of b:
\[ \| \mathbf{b} \| = \sqrt{(-3)^2 + 6^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]
Step 3: Normalize the Vectors
Now, normalize the vectors:
- Normalized a:
\[ \hat{\mathbf{a}} = \frac{\mathbf{a}}{\|\mathbf{a}\|} = \frac{\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}}{3} = \frac{1}{3}\mathbf{i} + \frac{2}{3}\mathbf{j} - \frac{2}{3}\mathbf{k} \]
- Normalized b:
\[ \hat{\mathbf{b}} = \frac{\mathbf{b}}{\|\mathbf{b}\|} = \frac{-3\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}}{7} = -\frac{3}{7}\mathbf{i} + \frac{6}{7}\mathbf{j} + \frac{2}{7}\mathbf{k} \]
Step 4: Find the Angle Bisector Vector
The vector along the angle bisector \( \mathbf{v} \) is given by:
\[ \mathbf{v} = \|\mathbf{b}\| \hat{\mathbf{a}} + \|\mathbf{a}\| \hat{\mathbf{b}} \]
Substituting the values:
\[ \mathbf{v} = 7 \left(\frac{1}{3}\mathbf{i} + \frac{2}{3}\mathbf{j} - \frac{2}{3}\mathbf{k}\right) + 3 \left(-\frac{3}{7}\mathbf{i} + \frac{6}{7}\mathbf{j} + \frac{2}{7}\mathbf{k}\right) \]
Calculating gives:
\[ \mathbf{v} = \frac{7}{3}\mathbf{i} + \frac{14}{3}\mathbf{j} - \frac{14}{3}\mathbf{k} - \frac{9}{7}\mathbf{i} + \frac{18}{7}\mathbf{j} + \frac{