Let `u=2i-j+3k" and "a=4i-j+2k`. The vector component of u orthogonal to a is

Options

A. `(1//7)(20i-5j+10k)`
B. `(1//7)(4i+24j+4k)`
C. `(1//7)(11i+2j+6k)`
D. `(-1//7)(6i+2j-11k)`

Answer

To find the vector component of \( \mathbf{u} \) orthogonal to \( \mathbf{a} \), we can follow these steps:

### Step 1: Identify the given vectors
We have:
\[ \mathbf{u} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k} \]
\[ \mathbf{a} = 4\mathbf{i} - \mathbf{j} + 2\mathbf{k} \]

### Step 2: Calculate the dot product \( \mathbf{u} \cdot \mathbf{a} \)
The dot product is calculated as follows:
\[ \mathbf{u} \cdot \mathbf{a} = (2)(4) + (-1)(-1) + (3)(2) \]
Calculating each term:
\[ = 8 + 1 + 6 = 15 \]

### Step 3: Calculate the magnitude of vector \( \mathbf{a} \)
The magnitude of \( \mathbf{a} \) is given by:
\[ |\mathbf{a}| = \sqrt{(4)^2 + (-1)^2 + (2)^2} = \sqrt{16 + 1 + 4} = \sqrt{21} \]

### Step 4: Find the projection of \( \mathbf{u} \) onto \( \mathbf{a} \)
The projection of \( \mathbf{u} \) onto \( \mathbf{a} \) is given by:
\[ \text{proj}_{\mathbf{a}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{a}}{|\mathbf{a}|^2} \mathbf{a} \]
Calculating \( |\mathbf{a}|^2 \):
\[ |\mathbf{a}|^2 = 21 \]
Now substituting the values:
\[ \text{proj}_{\mathbf{a}} \mathbf{u} = \frac{15}{21} \mathbf{a} = \frac{5}{7} \mathbf{a} \]
Substituting \( \mathbf{a} \):
\[ = \frac{5}{7} (4\mathbf{i} - \mathbf{j} + 2\mathbf{k}) = \left(\frac{20}{7}\mathbf{i} - \frac{5}{7}\mathbf{j} + \frac{10}{7}\mathbf{k}\right) \]

### Step 5: Calculate the vector component of \( \mathbf{u} \) orthogonal to \( \mathbf{a} \)
The vector component of \( \mathbf{u} \) orthogonal to \( \mathbf{a} \) is given by:
\[ \mathbf{u}_{\perp} = \mathbf{u} - \text{proj}_{\mathbf{a}} \mathbf{u} \]
Substituting the values:
\[ \mathbf{u}_{\perp} = (2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) - \left(\frac{20}{7}\mathbf{i} - \frac{5}{7}\mathbf{j} + \frac{10}{7}\mathbf{k}\right) \]
Calculating each component:
- For \( \mathbf{i} \):
\[ 2 - \frac{20}{7} = \frac{14}{7} - \frac{20}{7} = -\frac{6}{7} \]
- For \( \mathbf{j} \):
\[ -1 + \frac{5}{7} = -\frac{7}{7} + \frac{5}{7} = -\frac{2}{7} \]
- For \( \mathbf{k} \):
\[ 3 - \frac{10}{7} = \frac{21}{7} - \frac{10}{7} = \frac{11}{7} \]

Thus, the vector component of \( \mathbf{u} \) orthogonal to \( \mathbf{a} \) is:
\[ \mathbf{u}_{\perp} = -\frac{6}{7}\mathbf{i} - \frac{2}{7}\mathbf{j} + \frac{11}{7}\mathbf{k} \]

### Final Answer
\[ \mathbf{u}_{\perp} = -\frac{6}{7}\mathbf{i} - \frac{2}{7}\mathbf{j} + \frac{11}{7}\mathbf{k} \]