Give an example for a dot product of two vectors.
Understand the Problem
The question is asking for an example of how to compute the dot product of two vectors, which involves a specific mathematical operation where corresponding elements of the vectors are multiplied and then summed.
Answer
The dot product of the vectors $\mathbf{A}$ and $\mathbf{B}$ is $56$.
Answer for screen readers
The dot product of the vectors $\mathbf{A}$ and $\mathbf{B}$ is $56$.
Steps to Solve
- Define the Vectors
Let’s consider two vectors in $\mathbb{R}^3$ as an example:
$$ \mathbf{A} = \begin{pmatrix} 2 \ 3 \ 4 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 5 \ 6 \ 7 \end{pmatrix} $$
- Multiply Corresponding Elements
For the dot product, we need to multiply corresponding elements from both vectors:
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For the first element: $2 \cdot 5 = 10$
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For the second element: $3 \cdot 6 = 18$
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For the third element: $4 \cdot 7 = 28$
- Sum the Products
Now, we add all the products:
$$ 10 + 18 + 28 $$
Calculating that gives us:
$$ 10 + 18 = 28 $$
$$ 28 + 28 = 56 $$
- Final Result
Thus, the dot product of vectors $\mathbf{A}$ and $\mathbf{B}$ is:
$$ \mathbf{A} \cdot \mathbf{B} = 56 $$
The dot product of the vectors $\mathbf{A}$ and $\mathbf{B}$ is $56$.
More Information
The dot product is a fundamental operation in linear algebra and has applications in physics, computer graphics, and machine learning. It measures how similar two vectors are, with results that can indicate the angle between them.
Tips
- Forgetting to sum the products: Ensure you add all the multiplied values at the end.
- Mixing up element positions: Double-check that you correctly multiply corresponding elements from each vector.
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