Projection of `b = 2i + 3j -2k` in the direction of vector `a = i+2j+3k` is ........ And the vector determined by the projection is ............

Answer

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

### Step 1: Identify the vectors
Given:
- **b** = 2i + 3j - 2k
- **a** = i + 2j + 3k

### Step 2: Find the magnitude of vector **a**
The magnitude of vector **a** is calculated using the formula:
\[ |a| = \sqrt{a_x^2 + a_y^2 + a_z^2} \]
Where \( a_x, a_y, a_z \) are the components of vector **a**.

Calculating:
\[ |a| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 3: Calculate the unit vector in the direction of **a**
The unit vector **u** in the direction of **a** is given by:
\[ u = \frac{a}{|a|} \]
Calculating:
\[ u = \frac{1}{\sqrt{14}}i + \frac{2}{\sqrt{14}}j + \frac{3}{\sqrt{14}}k \]

### Step 4: Find the dot product of **b** and **a**
The dot product \( b \cdot a \) is calculated as follows:
\[ b \cdot a = (2)(1) + (3)(2) + (-2)(3) = 2 + 6 - 6 = 2 \]

### Step 5: Calculate the projection of **b** onto **a**
The projection of **b** onto **a** is given by the formula:
\[ \text{proj}_{a}(b) = \frac{b \cdot a}{|a|^2} a \]
Calculating \( |a|^2 \):
\[ |a|^2 = 14 \]
Now substituting the values:
\[ \text{proj}_{a}(b) = \frac{2}{14} a = \frac{1}{7} a \]

### Step 6: Substitute the vector **a** into the projection formula
Now substituting the vector **a**:
\[ \text{proj}_{a}(b) = \frac{1}{7} (i + 2j + 3k) = \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \]

### Final Answers
1. The scalar projection of **b** in the direction of **a** is \( \frac{2}{\sqrt{14}} \).
2. The vector determined by the projection is \( \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \).

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