In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j

Verified step by step guidance

1. Identify the components of the vectors \( \mathbf{v} = 5\mathbf{i} - 4\mathbf{j} \) and \( \mathbf{w} = -2\mathbf{i} - \mathbf{j} \).
2. Use the dot product formula: \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 \), where \( v_1 = 5, v_2 = -4, w_1 = -2, w_2 = -1 \).
3. Calculate \( v_1w_1 = 5 \times (-2) \).
4. Calculate \( v_2w_2 = (-4) \times (-1) \).
5. Add the results from the previous two steps to find \( \mathbf{v} \cdot \mathbf{w} \).

Key Concepts

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar. It is calculated by multiplying the corresponding components of the vectors and then summing those products. For vectors v = ai + bj and w = ci + dj, the dot product is given by v·w = ac + bd. This operation is essential for determining the angle between vectors and their relative direction.

Vector Components

Vectors in a two-dimensional space can be expressed in terms of their components along the x-axis and y-axis. For example, the vector v = 5i - 4j has a component of 5 in the x-direction and -4 in the y-direction. Understanding vector components is crucial for performing operations like the dot product, as it allows for the manipulation of the vectors in a coordinate system.

Scalar Result

The result of the dot product is a scalar quantity, which means it has magnitude but no direction. This scalar can provide information about the relationship between the two vectors, such as whether they are orthogonal (perpendicular) or the extent to which they point in the same direction. Recognizing the significance of the scalar result is important for interpreting the outcome of vector operations.