Problem 65 - Algebra and Trigonometry

Question

Explain how to find the dot product of two vectors.

Step-by-Step Solution

Answer

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 1: Understand the formula

The formula for the dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \cdot \vec{b} = |a| |b| cos θ\), where |a| and |b| are the magnitudes of the vectors and θ 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 \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\).

Step 2: 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.

Step 3: Sum the results

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