Conceptual Question For Vectors

[willmac]

For three vectors to add up to zero, what must be true of the magnitudes of the three vectors? Hint: Draw a diagram of the addition of three vectors to make a vector of zero length. Think of what geometrical relationship they must have.
a. All the magnitudes must be equal.
b. One of the magnitudes must be greater than the sum of the other two.
c. One of the magnitudes must be at least twice as great as each of the other two.
d. One of the magnitudes must be less than the difference of the other two.
e. The sum of two of the magnitudes must be greater than or equal to the third for all three vectors.

 

[SHISHKABOB]

hi willmac welcome to PF

What geometrical relationship do you think they must have?

 

[willmac]

I feel like they are going to make a right triangle, but I was thinking the answer was e. If the vectors go in opposite directions well at least one of the longest ones has to be negative and the other two have to add up in order to be equivalent to the longest one so when the vectors combine they make zero.

 

[SHISHKABOB]

Well, does it have to be a right triangle? If the sum of some set of vectors equals zero, that means that they end up where they started, basically.

But thinking about it as a triangle is the right way to go I think. Imagine some triangles, like a right triangle and an equilateral triangle, ones where you know the relationship between the lengths of their sides. Then apply those options to those triangles and see which rule works for all the triangles you can think of.

 

[asteriatic]

The correct answer is a. All the magnitudes must be equal. In order for three vectors to add up to zero, they must form a closed triangle with all sides being equal in length. This is because the sum of two sides of a triangle must be greater than or equal to the third side, and in this case, all three sides must be equal for the sum to be zero. This geometric relationship is known as the Triangle Inequality Theorem. Therefore, all the magnitudes must be equal for the vectors to add up to zero.