# . Find the scalar component of projection of the vector a = 2๐̂+ 3๐̂+ 5๐̂ on the vector b = 2๐̂- 2๐̂- �

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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.