Parallel Vectors: Proof & Like/Unlike Determination

nUnderstanding parallel vectors is crucial in various fields, including physics, engineering, and computer graphics. This article will guide you through the process of proving whether two vectors are parallel and determining if they point in the same (like) or opposite (unlike) directions. We will dissect two specific examples with detailed explanations and step-by-step solutions. Let's dive in!

Understanding Parallel Vectors

Before diving into specific examples, let's clarify what it means for two vectors to be parallel. Two vectors, say a and b, are considered parallel if one is a scalar multiple of the other. Mathematically, this can be expressed as:

$\vec{a} = k \vec{b}$

where k is a scalar. This scalar k determines whether the vectors are in the same direction (like) or opposite directions (unlike).

• If k > 0, the vectors are parallel and point in the same direction (like).
• If k < 0, the vectors are parallel and point in opposite directions (unlike).
• If k = 0, then one of the vectors is a zero vector.

Now, let's apply this concept to the given examples.

Example 1: Analyzing Vectors $\vec{a}=\binom{-6}{4}$ and $\vec{b}=\binom{-3}{2}$

In this section, we will meticulously analyze the vectors $\vec{a}=\binom{-6}{4}$ and $\vec{b}=\binom{-3}{2}$ to determine if they are parallel and, if so, whether they are like or unlike vectors. Our approach involves checking if one vector is a scalar multiple of the other.

Proving Parallelism

To prove that $\vec{a}$ and $\vec{b}$ are parallel, we need to find a scalar k such that $\vec{a} = k \vec{b}$. This means we need to check if:

$\binom{-6}{4} = k \binom{-3}{2}$

This vector equation can be broken down into two scalar equations:

-6 = -3*k

4 = 2*k

From the first equation, we can solve for k:

k = -6 / -3 = 2

From the second equation, we also solve for k:

k = 4 / 2 = 2

Since both equations give us the same value for k (k = 2), we can conclude that $\vec{a}$ is indeed a scalar multiple of $\vec{b}$. Therefore, the vectors are parallel.

Determining Direction (Like or Unlike)

Now that we've established that the vectors are parallel, we need to determine whether they point in the same direction (like) or opposite directions (unlike). This is determined by the sign of the scalar k.

In our case, k = 2, which is a positive number. Therefore, the vectors $\vec{a}$ and $\vec{b}$ are parallel and point in the same direction. They are like vectors.

Summary for Example 1

• Vectors $\vec{a}=\binom{-6}{4}$ and $\vec{b}=\binom{-3}{2}$ are parallel.
• They are like vectors, pointing in the same direction.

Example 2: Analyzing Vectors $\overrightarrow{ PQ }=\binom{-2}{-5}$ and $\overrightarrow{ RS }=\binom{6}{15}$

In this section, we will analyze the vectors $\overrightarrow{ PQ }=\binom{-2}{-5}$ and $\overrightarrow{ RS }=\binom{6}{15}$ to determine if they are parallel and, if so, whether they are like or unlike vectors. We'll use the same method as before, checking for a scalar multiple relationship.

Proving Parallelism

To prove that $\overrightarrow{ PQ }$ and $\overrightarrow{ RS }$ are parallel, we need to find a scalar k such that $\overrightarrow{ PQ } = k \overrightarrow{ RS }$. This means we need to check if:

$\binom{-2}{-5} = k \binom{6}{15}$

This vector equation can be broken down into two scalar equations:

-2 = 6*k

-5 = 15*k

From the first equation, we can solve for k:

k = -2 / 6 = -1/3

From the second equation, we also solve for k:

k = -5 / 15 = -1/3

Since both equations give us the same value for k (k = -1/3), we can conclude that $\overrightarrow{ PQ }$ is indeed a scalar multiple of $\overrightarrow{ RS }$. Therefore, the vectors are parallel.

Determining Direction (Like or Unlike)

Now that we've established that the vectors are parallel, we need to determine whether they point in the same direction or opposite directions. Again, this depends on the sign of the scalar k.

In this case, k = -1/3, which is a negative number. Therefore, the vectors $\overrightarrow{ PQ }$ and $\overrightarrow{ RS }$ are parallel and point in opposite directions. They are unlike vectors.

Summary for Example 2

• Vectors $\overrightarrow{ PQ }=\binom{-2}{-5}$ and $\overrightarrow{ RS }=\binom{6}{15}$ are parallel.
• They are unlike vectors, pointing in opposite directions.

Key Takeaways

• Two vectors are parallel if one is a scalar multiple of the other.
• If the scalar k is positive, the vectors are like (same direction).
• If the scalar k is negative, the vectors are unlike (opposite direction).

By understanding these concepts, you can confidently determine whether two vectors are parallel and analyze their directional relationship. Remember to always break down vector equations into scalar equations to solve for the scalar multiple. Practice with different examples to solidify your understanding.

Understanding vector parallelism is fundamental to many areas of mathematics and physics. By carefully applying the principles outlined above, you can confidently analyze and interpret vector relationships in a variety of contexts.

Further Learning

To deepen your understanding of vectors and their properties, explore resources like Khan Academy's vector algebra section at https://www.khanacademy.org/science/physics/linear-algebra/vectors-and-scalars/v/introduction-to-vectors-and-scalars.