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The dot product is a fundamental operation in vector algebra where two vectors are combined to produce a scalar, or a single number, rather than another vector. To compute the dot product between two vectors, you multiply corresponding components together and then add the results. The formula for vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]
In this exercise, when calculating \( \mathbf{u} \cdot \mathbf{v} \), you work out \( 3 \times 1 + (-2) \times 3 \) yielding \( -3 \). This shows how the dot product captures the projection of one vector onto another.

Vector addition is the process of combining two vectors to form a new vector. This is achieved by adding their corresponding components. For vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \), their addition results in:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1)\mathbf{i} + (u_2 + v_2)\mathbf{j} \]
In the exercise, combining \( \mathbf{v} \) and \( \mathbf{w} \) results in the vector \( 5 \mathbf{i} + 8 \mathbf{j} \). The addition is straightforward, as it involves no different operations aside from aligning like terms in their respective columns.

The distributive property in vector algebra denotes how the dot product interacts with vector addition. Specifically, the dot product is distributive over vector addition, meaning for any vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \),
\[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \]
This shows how the dot product can "distribute" across a sum of vectors. This property is evident in exercise parts (a) and (b), both yielding \( -1 \), illustrating that calculating the dot product with a sum of vectors provides the same result as summing the individual dot products.

Scalar multiplication involves multiplying a vector by a scalar (a single real number), which scales the vector without changing its direction unless the scalar is negative. Given a vector \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) and a scalar \( c \), the resultant vector is:
\[ c \mathbf{v} = (c v_1) \mathbf{i} + (c v_2) \mathbf{j} \]
In the exercise, when you calculate \( \mathbf{u}(\mathbf{v} \cdot \mathbf{w}) \), the scalar \( 19 \), resulting from the dot product \( \mathbf{v} \cdot \mathbf{w} \), scales the vector \( \mathbf{u} \) to produce \( 57 \mathbf{i} - 38 \mathbf{j} \). This illustrates how a vector's length can be adjusted by multiplying with a scalar.