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The dot product is a fundamental operation in vector algebra that helps us measure the integration between two vectors. Often denoted by the symbol \( \cdot \), the dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is computed as the sum of the products of their corresponding components. Mathematically, if \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), then the dot product is defined as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

This result is not a vector, but a scalar, which encapsulates how "in line" the two vectors are. A crucial aspect of the dot product is that it is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \). Here in our projection task, we calculate the dot product \( \mathbf{w} \cdot \mathbf{u} \) to find how much of \( \mathbf{w} \) is pointing in the direction of \( \mathbf{u} \). By summing \( 3 + 10 - 3 \), the dot product yields a value of 10 which acts as a crucial component in finding the projection.

Scalar multiplication involves multiplying a vector by a scalar (a real number), which results in a new vector. This operation is critical when scaling the vector to reflect a certain property or measurement, such as in projection where we adjust the length of one vector along another. For example, if we have a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) and a scalar \( x \), then the product of this scalar and the vector is given by:

• \( x\mathbf{u} = x(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = xa\mathbf{i} + xb\mathbf{j} + xc\mathbf{k} \)

In our solution, once we calculate the dot product and obtain the scalar \( \frac{5}{7} \), we use scalar multiplication to scale vector \( \mathbf{u} \). This operation results in each component of \( \mathbf{u} \) being multiplied by \( \frac{5}{7} \), giving us the new vector \( \frac{15}{7}\mathbf{i} + \frac{10}{7}\mathbf{j} + \frac{5}{7}\mathbf{k} \). This scaled vector represents how much of \( \mathbf{w} \) is aligned in the direction of \( \mathbf{u} \).

Understanding vector components is fundamental when working with vectors and projections. Every vector in three-dimensional space can be broken down into parts corresponding to the coordinate axes, typically represented as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) for the x, y, and z-axes, respectively. For a vector \( \mathbf{v} \), its components are essentially its shadow or impact along these axes:

• \( \mathbf{v} = v_x\mathbf{i} + v_y\mathbf{j} + v_z\mathbf{k} \)

These components facilitate vector operations like addition, scaling, and especially projection. By analyzing vectors through their components, we get a clearer picture of how to manipulate them in calculations. In the given problem, vectors \( \mathbf{u} \) and \( \mathbf{w} \) are broken down into their components, allowing us to perform the dot product and scalar multiplication accurately. This breakdown enables each vector to act effectively within operations or formulae, such as the projection where specific components are addressed individually.