If `a = 2i+j+2k, b=5i-3j+k`, then orthogonal projection vector of a and b is :

Options

A. `3i-3j+k`
B. `9(5i-3j+k)`
C. `(5i-3j+k)/(35)`
D. `(9(5i-3j+k))/(35)`

Answer

To find the orthogonal projection vector of vector **a** onto vector **b**, we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k} \]
\[ \mathbf{b} = 5\mathbf{i} - 3\mathbf{j} + \mathbf{k} \]

### Step 2: Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\)
The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (2)(5) + (1)(-3) + (2)(1) \]
Calculating each term:
\[ = 10 - 3 + 2 = 9 \]

### Step 3: Calculate the magnitude of vector \(\mathbf{b}\)
The magnitude of vector \(\mathbf{b}\) is given by:
\[ |\mathbf{b}| = \sqrt{(5)^2 + (-3)^2 + (1)^2} = \sqrt{25 + 9 + 1} = \sqrt{35} \]

### Step 4: Calculate the unit vector of \(\mathbf{b}\)
The unit vector \(\hat{\mathbf{b}}\) in the direction of \(\mathbf{b}\) is:
\[ \hat{\mathbf{b}} = \frac{\mathbf{b}}{|\mathbf{b}|} = \frac{5\mathbf{i} - 3\mathbf{j} + \mathbf{k}}{\sqrt{35}} \]

### Step 5: Calculate the orthogonal projection of \(\mathbf{a}\) onto \(\mathbf{b}\)
The formula for the orthogonal projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\) is:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \right) \mathbf{b} \]
Substituting the values we calculated:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{9}{35} \right) \mathbf{b} \]
Now substituting \(\mathbf{b}\):
\[ = \frac{9}{35} (5\mathbf{i} - 3\mathbf{j} + \mathbf{k}) \]
Distributing \(\frac{9}{35}\):
\[ = \frac{9 \cdot 5}{35} \mathbf{i} - \frac{9 \cdot 3}{35} \mathbf{j} + \frac{9 \cdot 1}{35} \mathbf{k} \]
\[ = \frac{45}{35} \mathbf{i} - \frac{27}{35} \mathbf{j} + \frac{9}{35} \mathbf{k} \]

### Final Answer:
Thus, the orthogonal projection vector of \(\mathbf{a}\) onto \(\mathbf{b}\) is:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{45}{35} \mathbf{i} - \frac{27}{35} \mathbf{j} + \frac{9}{35} \mathbf{k} \]

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