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Chapter 7: Problem 65

Explain how to find the dot product of two vectors.

Short Answer

Expert verified
The dot product of two vectors is calculated using the Cartesian component formula as follows: \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\), where \(a_1, a_2\) and \(b_1, b_2\) are the components of vectors \(\vec{a}\) and \(\vec{b}\) respectively. Multiply the corresponding components and sum the results to get the dot product.

Step by step solution

01

Understand the formula

The formula for the dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \cdot \vec{b} = |a| |b| cos 胃\), where |a| and |b| are the magnitudes of the vectors and 胃 is the angle between the vectors. However, when vectors are represented in Cartesian coordinates, i.e., \(\vec{a} = a_1\hat{i} + a_2\hat{j}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j}\), the dot product can be calculated using their individual components as \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\).
02

Multiply corresponding components

Firstly, multiply the respective i components (\(a_1\) and \(b_1\)) and the respective j components (\(a_2\) and \(b_2\)) of the vectors. This would give two scalar values.
03

Sum the results

Add the results of the multiplication from step 2 together. This results in the dot product of vectors \(\vec{a}\) and \(\vec{b}\).

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