Determining Vectors Collinear With (2 -3)

In linear algebra, understanding the concept of collinearity is crucial, especially when dealing with vectors. Collinear vectors are vectors that lie on the same line or parallel lines. This means they are scalar multiples of each other. In this article, we will explore how to determine which vector is collinear with the given vector (2, -3) from the options provided: A) [-4, 6], B) [12, 18], C) [2, -6], and D) [-2, -3]. We'll delve into the mathematical principles behind collinearity and apply them to solve this specific problem.

Understanding Collinear Vectors

Collinear vectors, at their core, are vectors that are scalar multiples of one another. This means that one vector can be obtained by multiplying the other vector by a scalar (a real number). Mathematically, if vector b is collinear with vector a, then there exists a scalar k such that b = ka. This relationship is the key to identifying collinear vectors. Visualizing this, collinear vectors either lie on the same line or are parallel. They can point in the same direction (if k is positive) or in opposite directions (if k is negative), but their directions are fundamentally related.

To determine if two vectors are collinear, we essentially check if their components are proportional. For two-dimensional vectors, this involves comparing the ratios of their corresponding components. If the ratios are equal, the vectors are collinear. For example, if we have vectors a = (a₁, a₂) and b = (b₁, b₂), they are collinear if b₁/a₁ = b₂/a₂. This proportionality ensures that one vector is simply a scaled version of the other. This concept extends to higher dimensions as well, where the proportionality must hold for all corresponding components.

The significance of understanding collinearity extends beyond theoretical mathematics. In physics, collinear vectors can represent forces acting along the same line, simplifying the analysis of their combined effect. In computer graphics, collinearity is essential in determining if points lie on the same line, which is crucial for rendering lines and shapes. Furthermore, in navigation and mapping, understanding collinearity helps in determining if routes are parallel or lie on the same path. Therefore, grasping the concept of collinearity is not just an academic exercise but a valuable tool in various practical applications.

Analyzing the Given Vector (2, -3)

Our main task is to identify which of the given vectors is collinear with the vector (2, -3). To achieve this, we need to understand the characteristics of vectors that are collinear with (2, -3). Any vector collinear with (2, -3) must be a scalar multiple of it. This means we can obtain a collinear vector by multiplying (2, -3) by any scalar k. The resulting vector will have the form (2k, -3k). This understanding provides a clear path for checking the given options. We are essentially looking for vectors whose components are in the same proportion as (2, -3), meaning the ratio of their x-components and y-components should be consistent.

The vector (2, -3) itself has a specific direction and magnitude. The direction can be visualized as a line in the two-dimensional plane, and any vector collinear with it will lie along the same line or a parallel line. The magnitude is the length of the vector, which is not directly relevant to collinearity but becomes important when considering vector operations like addition or subtraction. When we multiply (2, -3) by a positive scalar, we obtain a vector in the same direction but with a scaled magnitude. Multiplying by a negative scalar results in a vector in the opposite direction, but still collinear. The sign of the scalar determines whether the collinear vector points in the same or opposite direction as (2, -3).

Understanding the properties of the vector (2, -3) is crucial for the problem at hand. It serves as the reference against which we will compare the other vectors. The components 2 and -3 define its direction, and any change in these components' proportion will result in a non-collinear vector. Therefore, our analysis will focus on checking if the given options maintain this proportion when compared to (2, -3). This approach simplifies the problem to checking for scalar multiples, making it a straightforward application of the definition of collinearity.

Evaluating Option A: [-4, 6]

Now, let's examine Option A: [-4, 6] to determine if it is collinear with the vector (2, -3). To do this, we need to check if there exists a scalar k such that [-4, 6] = k**(2, -3)**. This means we need to find a k that satisfies both -4 = 2k and 6 = -3k. Solving the first equation, -4 = 2k, we get k = -2. Now, let's check if this value of k also satisfies the second equation: 6 = -3k. Substituting k = -2, we get 6 = -3*(-2), which simplifies to 6 = 6. Since both equations are satisfied with the same value of k, we can conclude that [-4, 6] is indeed a scalar multiple of (2, -3).

The scalar k = -2 indicates that the vector [-4, 6] is twice the magnitude of (2, -3) and points in the opposite direction. The negative sign is crucial here, as it signifies that the vectors are anti-parallel, which is still a form of collinearity. Visually, the vector [-4, 6] lies on the same line as (2, -3) but points in the opposite direction. This satisfies the condition for collinearity, as the vectors are on the same line, regardless of their direction.

Therefore, based on our analysis, Option A [-4, 6] is collinear with the vector (2, -3). This conclusion is reached by demonstrating that a scalar k exists that relates the two vectors, confirming that they are scalar multiples of each other. This rigorous check ensures that we have correctly applied the definition of collinearity and can confidently consider Option A as a valid solution. In the context of the given question, this means we have identified one potential answer, and we will proceed to evaluate the remaining options to ensure we identify all collinear vectors.

Evaluating Option B: [12, 18]

Next, we will evaluate Option B: [12, 18] to determine if it is collinear with the vector (2, -3). Following the same procedure as before, we need to check if there exists a scalar k such that [12, 18] = k**(2, -3)**. This gives us two equations to solve: 12 = 2k and 18 = -3k. Solving the first equation, 12 = 2k, we find k = 6. Now, we need to verify if this value of k also satisfies the second equation: 18 = -3k. Substituting k = 6, we get 18 = -3*(6), which simplifies to 18 = -18. This is clearly a contradiction, as 18 does not equal -18.

Since we could not find a single value of k that satisfies both equations, we can conclude that [12, 18] is not a scalar multiple of (2, -3). This means that the vectors are not collinear. The components of [12, 18] do not maintain the same proportion as the components of (2, -3). Specifically, the ratio of the x-components (12/2) is 6, while the ratio of the y-components (18/-3) is -6. These ratios are not equal, further confirming the non-collinearity of the vectors.

Visually, the vector [12, 18] does not lie on the same line as (2, -3) or a parallel line. It has a different direction, and no scaling factor can transform (2, -3) into [12, 18]. This geometric interpretation reinforces our algebraic conclusion. Therefore, Option B [12, 18] is not collinear with the vector (2, -3). This eliminates Option B as a potential solution to the problem.

Evaluating Option C: [2, -6]

Now, let’s consider Option C: [2, -6] and assess whether it is collinear with the vector (2, -3). To determine collinearity, we need to check if there exists a scalar k such that [2, -6] = k**(2, -3)**. This gives us the equations 2 = 2k and -6 = -3k. Solving the first equation, 2 = 2k, we find k = 1. Next, we verify if this value of k satisfies the second equation: -6 = -3k. Substituting k = 1, we get -6 = -3*(1), which simplifies to -6 = -3. This is a contradiction, indicating that k = 1 does not satisfy both equations.

Since no single value of k can satisfy both equations, we conclude that [2, -6] is not a scalar multiple of (2, -3). Therefore, the vectors are not collinear. The ratios of the corresponding components are not equal: the ratio of the x-components (2/2) is 1, while the ratio of the y-components (-6/-3) is 2. The inequality of these ratios confirms the non-collinearity.

In a visual context, the vector [2, -6] does not lie on the same line or a parallel line to (2, -3). Its direction is different, and no scalar multiplication can transform (2, -3) into [2, -6]. This visual representation supports our algebraic finding that the vectors are not collinear. Thus, Option C [2, -6] is not collinear with the vector (2, -3), and we can eliminate it from the possible solutions.

Evaluating Option D: [-2, -3]

Finally, we evaluate Option D: [-2, -3] to see if it is collinear with the vector (2, -3). We need to determine if there exists a scalar k such that [-2, -3] = k**(2, -3)**. This gives us the equations -2 = 2k and -3 = -3k. Solving the first equation, -2 = 2k, we find k = -1. Now, we need to check if this value of k satisfies the second equation: -3 = -3k. Substituting k = -1, we get -3 = -3*(-1), which simplifies to -3 = 3. This is a contradiction, as -3 does not equal 3.

Since we cannot find a single value of k that satisfies both equations, we conclude that [-2, -3] is not a scalar multiple of (2, -3). Hence, the vectors are not collinear. The ratios of the corresponding components are not equal: the ratio of the x-components (-2/2) is -1, while the ratio of the y-components (-3/-3) is 1. These unequal ratios confirm that the vectors are not collinear.

Visually, the vector [-2, -3] does not lie on the same line as (2, -3) or a parallel line. Its direction is significantly different, and no scalar multiplication can transform (2, -3) into [-2, -3]. This visual interpretation aligns with our algebraic determination that the vectors are not collinear. Therefore, Option D [-2, -3] is not collinear with the vector (2, -3), and we can eliminate it from the possible answers.

Conclusion: Identifying the Collinear Vector

After evaluating all the options, we can now conclude which vector is collinear with (2, -3). Our analysis involved checking each option to see if it was a scalar multiple of the given vector. We found that Option A: [-4, 6] is the only vector that satisfies this condition. By setting [-4, 6] = k**(2, -3)**, we determined that k = -2, which satisfies both component equations. This confirms that [-4, 6] is indeed collinear with (2, -3).

The other options, [12, 18], [2, -6], and [-2, -3], did not meet the criteria for collinearity. In each case, we could not find a single scalar k that would make the given vector a multiple of (2, -3). This highlights the importance of the scalar multiple relationship in defining collinearity. Understanding this relationship allows us to efficiently identify vectors that lie on the same line or parallel lines.

In summary, the vector [-4, 6] is collinear with (2, -3) because it is a scalar multiple of (2, -3), specifically -2 times the original vector. This conclusion is supported by both algebraic and visual interpretations of collinearity. This exercise demonstrates the practical application of linear algebra concepts in identifying relationships between vectors, a fundamental skill in various mathematical and scientific fields.