Questions: How much is (3i+2j) · (7i-3j)?

Transcript text: The dot product of two vectors $\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}$ and $\mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}$ is defined as $\mathbf{u} \cdot \mathbf{v}=a_{1} \cdot a_{2}+b_{1} \cdot b_{2}$. Question: How much is $(3 \mathbf{i}+2 \mathbf{j}) \cdot(7 \mathbf{i}-3 \mathbf{j})$ ?

Solution

Solution Steps

To find the dot product of two vectors, we multiply the corresponding components of the vectors and then sum these products. For the vectors $ \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} $ and $ \mathbf{v} = 7\mathbf{i} - 3\mathbf{j} $, we calculate the dot product as follows: multiply the $ \mathbf{i} $ components (3 and 7) and the $ \mathbf{j} $ components (2 and -3), then add the results.

Step 1: Define the Vectors

We are given two vectors: \[ \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} \] \[ \mathbf{v} = 7\mathbf{i} - 3\mathbf{j} \]

Step 2: Calculate the Dot Product

The dot product $ \mathbf{u} \cdot \mathbf{v} $ is calculated using the formula: \[ \mathbf{u} \cdot \mathbf{v} = a_{1} \cdot a_{2} + b_{1} \cdot b_{2} \] Substituting the components: \[ \mathbf{u} \cdot \mathbf{v} = (3)(7) + (2)(-3) \]

Step 3: Perform the Multiplication

Calculating each term: \[ (3)(7) = 21 \] \[ (2)(-3) = -6 \]

Step 4: Sum the Results

Now, we sum the results of the multiplications: \[ \mathbf{u} \cdot \mathbf{v} = 21 - 6 = 15 \]