Magnitude of vector sums | Vectors | Precalculus | Khan Academy

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.

Transcript

Sal:Let's say that we have three vectors, vectors A, B and C, and we know that vector A plus vector B is equal to vector C. Now given this I have some interesting questions. Can you construct a scenario where the magnitude of vector C is equal to the magnitude of vector A plus the magnitude of vector B? And can you also, using potentially different... Read More

Key Insights

• 💁 Vector addition forms a triangle, with the resulting vector always being the third side.
• 🔙 The magnitude of vector C can only be equal to the sum of the magnitudes of vector A and vector B if they are 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 if they are not in the same direction.
• 🍹 Changing the direction of vector B will not make the magnitude of vector C greater than the sum of the magnitudes of vector A and vector B.
• 🔙 The largest possible magnitude of vector C is the sum of the magnitudes of vector A and vector B, which only occurs when A and B are in the same direction.
• 🙃 One side of a triangle can never be longer than the sum of the other two sides.
• 🔙 The scenario where the magnitude of vector C is less than the sum of the magnitudes of vector A and vector B is typical when the vectors are not 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.