What is the dot product of two parallel vectors?
Understand the Problem
The question is asking for the calculation of the dot product between two parallel vectors. The dot product can be computed using the formula involving the cosine of the angle between the vectors, which is zero for parallel vectors. This will help us find the result.
Answer
The dot product is $|\mathbf{A}| |\mathbf{B}|$ or $-|\mathbf{A}| |\mathbf{B}|$ depending on their directions.
Answer for screen readers
The dot product of two parallel vectors is either $|\mathbf{A}| |\mathbf{B}|$ (if they point in the same direction) or $-|\mathbf{A}| |\mathbf{B}|$ (if they point in opposite directions).
Steps to Solve
- Identify the formula for dot product
The formula to find the dot product $\mathbf{A} \cdot \mathbf{B}$ of two vectors can be expressed as: $$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) $$ where $|\mathbf{A}|$ is the magnitude of vector $\mathbf{A}$, $|\mathbf{B}|$ is the magnitude of vector $\mathbf{B}$, and $\theta$ is the angle between the two vectors.
- Determine the angle for parallel vectors
For parallel vectors, the angle $\theta$ is either $0$ degrees or $180$ degrees. In both cases, the cosine value is: $$ \cos(0) = 1 \quad \text{and} \quad \cos(180) = -1 $$
- Calculate the dot product using cosine
If the vectors are parallel and pointing in the same direction (0 degrees): $$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cdot 1 $$
If they are parallel but pointing in opposite directions (180 degrees): $$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cdot (-1) $$
- Final conclusion on the result
Thus, for parallel vectors, the dot product is the product of their magnitudes, with a sign depending on their directions.
The dot product of two parallel vectors is either $|\mathbf{A}| |\mathbf{B}|$ (if they point in the same direction) or $-|\mathbf{A}| |\mathbf{B}|$ (if they point in opposite directions).
More Information
The dot product is a useful operation in physics and engineering as it helps derive information such as work done when force is applied along a certain direction.
Tips
- Confusing the angle between vectors; remember that parallel vectors have an angle of $0$ or $180$ degrees.
- Not considering the directions of the vectors when calculating the dot product.
