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

Answer: Yes, it is possible to add three equal-length vectors and obtain a vector sum of zero by arranging them in a way that they form an equilateral triangle.

Step-by-step solution

Understanding vector addition

Vector addition can be thought of as the combination of the magnitudes and directions of two or more vectors. Geometrically, the sum of two vectors can be found by placing the tail of one vector at the head of the other, and then finding the closing side of the triangle formed. This is called the Triangle Law of vector addition.

Analyzing possible arrangements of three equal-length vectors

We need to find an arrangement of three equal-length vectors such that their sum is zero. We can start by considering the geometric approach to vector addition and thinking about how we can arrange the three vectors to form a closed triangle.

Forming a closed triangle with three equal-length vectors

To form a closed triangle, the three vectors must create an equilateral triangle. In an equilateral triangle, all sides have equal length, and all angles are equal, with each angle measuring 60 degrees. If we can arrange the three equal-length vectors to form an equilateral triangle, then the sum of the three vectors will be zero.

Sketching the arrangement of the three vectors

Let's sketch an arrangement for the three equal-length vectors A, B, and C, while forming an equilateral triangle. 1. Draw vector A in any direction. 2. Draw vector B from the head of vector A, making a 60-degree angle with A. 3. Draw vector C from the head of vector B, making a 60-degree angle with B and directing towards the tail of vector A. Upon completing the drawing, we can observe that when we combine A, B, and C, they form a closed equilateral triangle which leads back to the starting position.

Conclusion

It is possible to add three equal-length vectors and obtain a vector sum of zero. This can be achieved by arranging the three vectors in a way that they form an equilateral triangle.