Magnitude of vector sums  Vectors  Precalculus  Khan Academy  Summary and Q&A
TL;DR
Vector addition always forms a triangle, and the magnitude of the sum (vector C) will never be greater than the sum of the magnitudes of A and B unless A and B are in the same direction.
Questions & Answers
Q: Can you provide an example of vector addition that results in the magnitude of vector C being equal to the sum of the magnitudes of vector A and vector B?
Yes, the only way for the magnitude of vector C to be equal to the sum of the magnitudes of vector A and vector B is if A and B are in the exact same direction. This creates a scenario where the triangle formed by A, B, and C has sides of equal length.
Q: Is it possible for the magnitude of vector C to be greater than the sum of the magnitudes of vector A and vector B?
No, it is not possible. The triangle formed by A, B, and C will always have one side (vector C) that is shorter than the sum of the other two sides (magnitudes of vector A and vector B). To make the magnitude of vector C greater, the direction of vector B would need to align exactly with vector A, which is not possible.
Q: What happens if vector A and vector B are not in the same direction?
If vector A and vector B are not in the same direction, the magnitude of vector C will always be less than the sum of the magnitudes of vector A and vector B. This is because vector C will always be the shortest side of the triangle formed by A, B, and C.
Q: Can you provide an example of vector addition where the magnitude of vector C is less than the sum of the magnitudes of vector A and vector B?
Yes, if vector A and vector B are not in the same direction, the magnitude of vector C will be less than the sum of the magnitudes of vector A and vector B. This can be visualized by drawing A and B in different directions and then adding them together. The resulting vector C will always be shorter than the sum of the magnitudes of A and B.
Summary & Key Takeaways

Vector addition, where A plus B equals C, always forms a triangle.

The magnitude of vector C can only be equal to the sum of the magnitudes of vector A and vector B if A and B are in the same direction.

If A and B are not in the same direction, the magnitude of vector C will always be less than the sum of the magnitudes of vector A and vector B.