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Chapter 6: Problem 65

Explain how to find the dot product of two vectors.

Short Answer

Expert verified
The dot product of vectors \( \mathbf{v} = \langle v_1, v_2 \rangle \) and \( \mathbf{w} = \langle w_1, w_2 \rangle \) is calculated by multiplying the corresponding components and adding those products, represented by the formula \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 \).

Step by step solution

01

Express the vectors

If not already done, express the vectors in component form. A vector \( \mathbf{v} \) in two dimensions is often written as \( \mathbf{v} = \langle v_1, v_2 \rangle \), where \( v_1 \) and \( v_2 \) are the components of \( \mathbf{v} \). Similarly, a vector \( \mathbf{w} \) may be expressed as \( \mathbf{w} = \langle w_1, w_2 \rangle \).
02

Multiply the corresponding components

Multiply the corresponding components of the two vectors. That is, multiply the first component of vector \( \mathbf{v} \) with the first component of vector \( \mathbf{w} \), and likewise for the second components. This gives you \( v_1w_1 \) and \( v_2w_2 \) respectively.
03

Add the products

Add the products of the corresponding components together to compute the dot product. This should be done as follows: \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 \). This final value is the dot product of vectors \( \mathbf{v} \) and \( \mathbf{w} \).

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