In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w

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Recall that the dot product of two vectors \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) and \( \mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} \) is given by the formula: \[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]

Identify the components of the given vectors: For \( \mathbf{v} = -5 \mathbf{i} + 2 \mathbf{j} \), we have \( v_1 = -5 \) and \( v_2 = 2 \). For \( \mathbf{w} = 2 \mathbf{i} - 4 \mathbf{j} \), we have \( w_1 = 2 \) and \( w_2 = -4 \).

Substitute the components into the dot product formula: \[ \mathbf{v} \cdot \mathbf{w} = (-5)(2) + (2)(-4) \]

Simplify the expression by performing the multiplications and then adding the results: \[ (-5)(2) + (2)(-4) = -10 + (-8) \]

Add the two products to find the dot product value: \[ -10 + (-8) = -18 \]

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주요 개념

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Dot Product of Vectors

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components and summing the results, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2.

Vector Components and Notation

Vectors in two dimensions are expressed in terms of unit vectors i and j, representing the x and y directions respectively. Understanding how to identify and manipulate these components is essential for performing operations like the dot product.

Geometric Interpretation of the Dot Product

The dot product relates to the angle between two vectors: v ⋅ w = |v||w|cosθ, where θ is the angle between v and w. This relationship helps in finding the angle or understanding the projection of one vector onto another.