. Find the scalar component of projection of the vector a = 2𝑖̂+ 3𝑗̂+ 5𝑘̂ on the vector b = 2𝑖̂- 2𝑗̂- �

The scalar component of the projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by the formula:

\[ \text{Scalar Projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \]

Here, \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \|\mathbf{b}\| \) is the magnitude of vector \( \mathbf{b} \).

Let's calculate it step by step:

Given vectors:

\[ \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \]

\[ \mathbf{b} = 2\mathbf{i} - 2\mathbf{j} - 3\mathbf{k} \]

1. Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \):

\[ \mathbf{a} \cdot \mathbf{b} = (2 \cdot 2) + (3 \cdot -2) + (5 \cdot -3) \]

2. Calculate the magnitude \( \|\mathbf{b}\| \):

\[ \|\mathbf{b}\| = \sqrt{2^2 + (-2)^2 + (-3)^2} \]

3. Substitute the values into the formula for the scalar projection:

\[ \text{Scalar Projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \]

Now, calculate the result.