How to find the scalar product of two vectors?
Understand the Problem
The question is asking for the method or formula to calculate the scalar product, also known as the dot product, of two vectors. It typically involves multiplying the corresponding components of the vectors and summing those products.
Answer
The dot product of vectors $\mathbf{A} = (A_1, A_2, A_3)$ and $\mathbf{B} = (B_1, B_2, B_3)$ is given by: $$ \mathbf{A} \cdot \mathbf{B} = A_1 \cdot B_1 + A_2 \cdot B_2 + A_3 \cdot B_3 $$
Answer for screen readers
The scalar product (dot product) of two vectors $\mathbf{A} = (A_1, A_2, A_3)$ and $\mathbf{B} = (B_1, B_2, B_3)$ is calculated as: $$ \mathbf{A} \cdot \mathbf{B} = A_1 \cdot B_1 + A_2 \cdot B_2 + A_3 \cdot B_3 $$
Steps to Solve
- Define the Vectors
Let the two vectors be represented as $\mathbf{A} = (A_1, A_2, A_3)$ and $\mathbf{B} = (B_1, B_2, B_3)$. Here, $A_1$, $A_2$, and $A_3$ are the components of vector $\mathbf{A}$, and $B_1$, $B_2$, and $B_3$ are the components of vector $\mathbf{B}$.
- Multiply Corresponding Components
To find the scalar (dot) product, multiply each corresponding component of the vectors: $$ A_1 \cdot B_1, \quad A_2 \cdot B_2, \quad A_3 \cdot B_3 $$
- Sum the Products
Add the results from the previous step to get the scalar product: $$ \mathbf{A} \cdot \mathbf{B} = A_1 \cdot B_1 + A_2 \cdot B_2 + A_3 \cdot B_3 $$
- Final Formula
Thus, the formula for the dot product of the two vectors can be summarized as: $$ \mathbf{A} \cdot \mathbf{B} = \sum_{i=1}^{n} A_i \cdot B_i $$ where $n$ is the number of dimensions (or components) of the vectors.
The scalar product (dot product) of two vectors $\mathbf{A} = (A_1, A_2, A_3)$ and $\mathbf{B} = (B_1, B_2, B_3)$ is calculated as: $$ \mathbf{A} \cdot \mathbf{B} = A_1 \cdot B_1 + A_2 \cdot B_2 + A_3 \cdot B_3 $$
More Information
The dot product is a fundamental operation in vector algebra, commonly used in physics and engineering. It helps in determining the angle between two vectors and is crucial in concepts like projections.
Tips
- Not using the correct components of the vectors: Ensure that the correct components are matched when multiplying.
- Confusing the dot product with the cross product, which is a different operation.
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