Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.
u • v - u • w

검증된 단계별 안내

Recall that the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by the formula: \[ \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \]

Calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \) using the components of \( \mathbf{u} = \langle -2, 1 \rangle \) and \( \mathbf{v} = \langle 3, 4 \rangle \): \[ \mathbf{u} \cdot \mathbf{v} = (-2) \times 3 + 1 \times 4 \]

Calculate the dot product \( \mathbf{u} \cdot \mathbf{w} \) using the components of \( \mathbf{u} = \langle -2, 1 \rangle \) and \( \mathbf{w} = \langle -5, 12 \rangle \): \[ \mathbf{u} \cdot \mathbf{w} = (-2) \times (-5) + 1 \times 12 \]

Substitute the results from the two dot products into the expression \( \mathbf{u} \cdot \mathbf{v} - \mathbf{u} \cdot \mathbf{w} \) to get: \[ (\mathbf{u} \cdot \mathbf{v}) - (\mathbf{u} \cdot \mathbf{w}) \]

Simplify the expression by performing the arithmetic operations to find the final value of \( \mathbf{u} \cdot \mathbf{v} - \mathbf{u} \cdot \mathbf{w} \).

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주요 개념

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Dot Product of Vectors

The dot product is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. It is calculated by multiplying corresponding components and summing the results. For vectors u = 〈u1, u2〉 and v = 〈v1, v2〉, the dot product is u • v = u1*v1 + u2*v2.

Vector Components and Notation

Vectors in two dimensions are represented as ordered pairs 〈x, y〉, where x and y are components along the horizontal and vertical axes. Understanding this notation is essential for performing operations like addition, subtraction, and dot product on vectors.

Properties of the Dot Product

The dot product is distributive over vector addition and subtraction, meaning u • (v - w) = u • v - u • w. This property allows simplification of expressions involving multiple dot products, facilitating easier calculation and understanding of vector relationships.

Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.u • v - u • w