In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. 4(u ⋅ v)

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First, recall the formula for the dot product of two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), which is \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Identify the components of vectors \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \). Here, \( u_1 = 2, u_2 = -1, v_1 = 3, v_2 = 1 \).

Substitute these components into the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = (2)(3) + (-1)(1) \).

Calculate the expression inside the parentheses: \( 2 \times 3 + (-1) \times 1 \).

Finally, multiply the result of the dot product by 4 as specified: \( 4(\mathbf{u} \cdot \mathbf{v}) \).

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Dot Product

The dot product is a fundamental operation in vector algebra that takes two vectors and returns a scalar. It is calculated by multiplying the corresponding components of the vectors and summing the results. For vectors u = ai + bj and v = ci + dj, the dot product is given by u ⋅ v = ac + bd. This operation is crucial for determining the angle between vectors and for projecting one vector onto another.

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Scalar multiplication involves multiplying a vector by a scalar (a single number), which scales the vector's magnitude without changing its direction. If k is a scalar and v is a vector, then k * v results in a new vector whose length is k times that of v. This concept is essential when manipulating vectors in various operations, including scaling the result of the dot product.

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Vectors in a two-dimensional space can be expressed in terms of their components along the x-axis and y-axis. For example, a vector u = ai + bj has components a and b, where 'i' represents the unit vector in the x-direction and 'j' in the y-direction. Understanding vector components is vital for performing operations like the dot product, as it allows for the straightforward application of the formula using the individual components of the vectors.

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