In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

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1. Identify the components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \). For \( \mathbf{v} = -6\mathbf{i} - 5\mathbf{j} \), the components are \( v_1 = -6 \) and \( v_2 = -5 \). For \( \mathbf{w} = -10\mathbf{i} - 8\mathbf{j} \), the components are \( w_1 = -10 \) and \( w_2 = -8 \).
2. To find the dot product \( \mathbf{v} \cdot \mathbf{w} \), use the formula: \( \mathbf{v} \cdot \mathbf{w} = v_1 \cdot w_1 + v_2 \cdot w_2 \).
3. Substitute the components into the dot product formula: \( (-6) \cdot (-10) + (-5) \cdot (-8) \).
4. Calculate each term separately: \((-6) \cdot (-10)\) and \((-5) \cdot (-8)\).
5. Add the results of the two products 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 are represented in terms of their components along the coordinate axes. In the case of v = -6i - 5j, the components are -6 (along the x-axis) and -5 (along the y-axis). 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.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula |v| = √(a² + b²) for a vector v = ai + bj. This concept is important when calculating the dot product of a vector with itself, as it provides insight into the vector's size and is used in various applications, including physics and engineering.