Question

Is it possible to add three equal-length vectors and obtain a vector sum of zero? If so, sketch the arrangement of the three vectors. If not, explain why not.

Short answer

If so, sketch their arrangement. Answer: Yes, it is possible. The arrangement of the three equal-length vectors A, B, and C can be in the form of an equilateral triangle with angles of 120 degrees between each vector. When arranged in this manner, the sum of their x and y components is zero, indicating that their vector sum is zero.

Step-by-step solution

Understand the properties of vectors

Vectors have both magnitudes and directions. Two vectors will be equal if they have the same magnitude and direction. To find the sum of vectors, we can first break down each vector into its components along the x and y axes. The vector sum will be the sum of the components of each vector.

Set up an arrangement with three equal-length vectors

To begin, let's assign the equal-length vectors A, B, and C, with all having the same magnitude. We can arrange these vectors in a way that they form an equilateral triangle, each with an angle of 120 degrees with respect to each other.

Break down the vectors into components

Next, let's break down each vector into its x and y components. For vector A, which has a direction of 0 degrees (a horizontal vector), we have: A_x = 1 * magnitude A_y = 0 For vector B at 120 degrees, we have: B_x = -1/2 * magnitude B_y = (\sqrt{3}/2) * magnitude For vector C at 240 degrees, we have: C_x = -1/2 * magnitude C_y = -(\sqrt{3}/2) * magnitude

Calculate the sum of vector components

Now, let's add the x and y components of each vector A, B, and C: Sum_x = (1 * magnitude) + (-1/2 * magnitude) + (-1/2 * magnitude) = 0 Sum_y = (0) + (\sqrt{3}/2) * magnitude + (-\sqrt{3}/2) * magnitude = 0 Since the sum of the x and y components is 0, the vector sum is also 0.

Sketch the arrangement of the three vectors

To sketch the arrangement of the three equal-length vectors A, B, and C, simply draw an equilateral triangle with A starting from the origin and pointing to the right, B starting from the tip of A and pointing up at an angle of 120 degrees, and C starting from the tip of B and pointing down at an angle of 240 degrees. The tip of C will coincide with the origin, showing that their vector sum is indeed zero.

Key concepts

Vector Components

In physics and mathematics, vector components are the projections of a vector onto the axes of a coordinate system. For example, any vector in two-dimensional space can be represented by its components along the x-axis (horizontal) and y-axis (vertical). This decomposition into components is fundamental as it allows for simpler calculations concerning vector addition and subtraction.

It's done by multiplying the magnitude of the vector by the cosine of the angle it makes with the x-axis for the horizontal component, and by the sine of the angle for the vertical component. In an exercise where we have vectors with equal magnitudes, breaking them down into their components (\( A_x, A_y, B_x, B_y, C_x, C_y \)) enables us to analyze their contributions along each axis and helps students understand the effects of vector addition in a more granular fashion.

For example, if we have a horizontal vector A with a magnitude of 1, its x-component would be 1 (since the cosine of 0 degrees is 1) and its y-component would be 0 (since the sine of 0 degrees is 0). Recognizing these components helps us understand how vectors in different orientations contribute to the resultant vector—or in the case of our exercise, how they can cancel each other out.

Equilateral Triangle

Geometry in Vectors

An equilateral triangle is a perfect candidate to demonstrate the concept of vector sums due to its geometric properties: all sides are of equal length, and all interior angles are equal, each measuring 120 degrees. This symmetry is what allows us to arrange vectors in a way that their sum leads to a specific result, such as zero.

In the provided exercise, using an equilateral triangle arrangement benefits students in visualizing classically difficult vector concepts. By aligning the vectors tail-to-head in this shape, each vector effectively 'undoes' the progress of the previous one. It affords a vivid insight into how direction and magnitude work in unison to define vector quantities. Learning to sketch such arrangements sharpens a student's ability to predict the outcomes of vector addition, which is a critical skill in fields like physics and engineering.

Magnitude and Direction

Vectors are defined by two primary characteristics: magnitude and direction. The magnitude tells us about the size or length of the vector, while the direction informs us which way the vector is pointing. When adding vectors, both of these attributes must be considered to arrive at a correct result.

The exercise showcases that even if vectors have the same magnitude, their direction can lead to vastly different outcomes when they are combined. If vectors are arranged in such a way that they point back to the starting point—much like in our equilateral triangle example—their combined effect, or resultant vector, can be zero. Understanding that the vector sum is the combination of both these aspects—the total movement from the start point to the endpoint—forms the core of vector operations.

In practical applications, referring to the magnitude and direction could lead to solving complex problems involving forces, velocity, or even fields in physics, making it a cornerstone concept for students to grasp.