How to find a vector parallel to another vector?
Understand the Problem
The question is asking for the method to determine a vector that is parallel to a given vector. This typically involves understanding the properties of vectors and how to manipulate them mathematically to find a suitable parallel vector.
Answer
The parallel vector is $\begin{pmatrix} 6 \\ 8 \end{pmatrix}$ when $\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $k = 2$.
Answer for screen readers
The parallel vector to $\mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}$ with a scalar $k = 2$ is $\mathbf{u} = \begin{pmatrix} 6 \ 8 \end{pmatrix}$.
Steps to Solve
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Identify the Given Vector Start by noting the components of the given vector, denoted as $\mathbf{v} = \begin{pmatrix} v_1 \ v_2 \end{pmatrix}$.
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Understanding Parallel Vectors Parallel vectors have the same direction, meaning one is a scalar multiple of the other. Thus, if $\mathbf{u}$ is parallel to $\mathbf{v}$, we can express it as $\mathbf{u} = k \cdot \mathbf{v}$, where $k$ is any non-zero scalar.
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Choosing a Scalar Select a scalar $k$. For example, $k = 2$ will give us a vector that is twice as long but still parallel to $\mathbf{v}$.
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Calculating the Parallel Vector Now calculate the parallel vector using your chosen scalar. If we take $\mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}$, and $k = 2$, then: $$ \mathbf{u} = 2 \cdot \begin{pmatrix} 3 \ 4 \end{pmatrix} = \begin{pmatrix} 6 \ 8 \end{pmatrix} $$
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Result The result $\mathbf{u} = \begin{pmatrix} 6 \ 8 \end{pmatrix}$ is parallel to the original vector $\mathbf{v}$.
The parallel vector to $\mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}$ with a scalar $k = 2$ is $\mathbf{u} = \begin{pmatrix} 6 \ 8 \end{pmatrix}$.
More Information
The key concept of parallel vectors allows us to see that their direction remains consistent even when their magnitude changes. You can pick any scalar (positive or negative) to generate varied parallel vectors.
Tips
- Choosing Zero for the Scalar: Choosing $k = 0$ results in the zero vector, which does not have a defined direction and is not considered parallel to any non-zero vector.
- Forgetting Direction: Ensure that the direction is maintained even if you choose a negative scalar, resulting in a vector that is parallel but points in the opposite direction.
