If `a=i+j+k" and "b=i-j+2k` then the projection of a on b is given by

Options

A. `1/2(i-j+2k)`
B. `1/3(i+j+k)`
C. `1/3(i-j-k)`
D. `1/3(i-j+2k)`

Answer

To find the projection of vector **a** on vector **b**, we will use the formula for the projection of one vector onto another. The formula is given by:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]

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

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

### Step 3: Calculate the magnitude squared of vector \(\mathbf{b}\)
The magnitude squared of vector \(\mathbf{b}\) is given by:
\[ |\mathbf{b}|^2 = (1)^2 + (-1)^2 + (2)^2 \]
Calculating this:
\[ = 1 + 1 + 4 = 6 \]

### Step 4: Substitute into the projection formula
Now we substitute the values into the projection formula:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{2}{6} \mathbf{b} \]
This simplifies to:
\[ = \frac{1}{3} \mathbf{b} \]

### Step 5: Substitute \(\mathbf{b}\) back into the equation
Now we substitute \(\mathbf{b} = \mathbf{i} - \mathbf{j} + 2\mathbf{k}\):
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{1}{3} (\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \]
Distributing \(\frac{1}{3}\):
\[ = \frac{1}{3} \mathbf{i} - \frac{1}{3} \mathbf{j} + \frac{2}{3} \mathbf{k} \]

### Final Result
Thus, the projection of vector **a** on vector **b** is:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{1}{3} \mathbf{i} - \frac{1}{3} \mathbf{j} + \frac{2}{3} \mathbf{k} \]

---