|
Title: Can you find two vectors with different lengths that have a vector sum of zero? Post by: micike on Jan 27, 2013 What restrictions are required for three vectors to have a vector sum of zero?
Can you find a vector quantity that has magnitude of zero but components that are different from zero? Can the magnitude of a vector be less than the magnitude of any of its components? Title: Can you find two vectors with different lengths that have a vector sum of zero? Post by: jsu4574 on Jan 27, 2013 Content hidden
Title: Can you find two vectors with different lengths that have a vector sum of zero? Post by: jsu4574 on Jan 27, 2013 There are three questions here, and we answer them in order:
(1) For two vectors to add to the 0 vector, they have to have the same magnitude and have opposite directions, so it is impossible to have two vectors of different lengths add to 0. (2) For three vectors to have a vector sum of 0, it is necessary and sufficient that the kth components of each of these vectors must also add up to o, for all values of k. (3) Since the magnitude of a vecor is the square-root of the sum of the squares of its components, and sqaures are positive, we have the inequality that the magnitude of a vector is greater than or equal to the sum of the magnitudes of each of its components. This is an n-dimensional analogue of the Triangle Inequality. So, A. If a vector has magnitude 0, then so do each of its components, and B. The magnitude of a vector is always at least as great as the magnitude of any component. |