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

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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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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.

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Introduction to Dot Product

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.

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i & j Notation

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.

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Domanda del libro di testo

Find the area of each triangle ABC.

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