Thus, beginning and ends of two vectros and of their sum form a triangle or a line. Magnitudes of these 2 vectors and of their sum are lengths of respective sides of the triangle. If it is equal to the difference of magnitudes,than it is lesser than their sum,because magnitudes are positive numbers. Show Can the magnitude of the component vector be greater than the magnitude of that vector?The components of a vector can never have a magnitude greater than the vector itself. This can be seen by using Pythagorean’s Thereom. Can the magnitude of the resultant of two vectors be less than the magnitude of either of those two vectors? The magnitude of the resultant of two vectors cannot be less than the magnitude of either of those two vectors. If A-B = -0, then the vectors A and B have equal magnitudes and are directed in the same direction. The magnitude of a vector can only be zero if all of its components are zero. Is the magnitude of the sum of two vectors necessarily greater than the magnitude of each vector explain? The magnitude of the sum of two vectors is always less than the sum of the magnitudes of the two vectors. A vector’s component can never be larger than the magnitude of the vector. It is possible for a component of a vector to be zero if the vector itself is not zero. What is the magnitude of 2 vectors?Formulas for the magnitude of vectors in two and three dimensions in terms of their coordinates are derived in this page. For a two-dimensional vector a=(a1,a2), the formula for its magnitude is ∥a∥=√a21+a22. Can the magnitude of a resultant vector be less than the magnitude of any of its components explain?The magnitude of any resultant vector of two components vectors can not be smaller than any of its component vectors because the positive combination… Can the magnitude of a resultant vector be less than the magnitude of any of its components? Can the sum of two vectors of equal magnitude be equal to magnitude of either of the vectors? If the vectors are equal, then their sum will necessarily have a larger magnitude than either of them unless the vector is zero. What is the difference between vector and magnitude?vector: A directed quantity, one with both magnitude and direction; the between two points. magnitude: A number assigned to a vector indicating its length. What is the magnitude of vector?The magnitude of a vector is the length of the vector. The magnitude of the vector a is denoted as ∥a∥. For a two-dimensional vector a=(a1,a2), the formula for its magnitude is ∥a∥=√a21+a22.
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In $\mathbb{R}^n$, we have that your equality implies $$\langle a+b,a+b \rangle=(\Vert a\Vert+\Vert b \Vert)^2 ,$$ which, using the bilinearity of the inner product, yields $$\Vert a \Vert^2+\Vert b \Vert^2 +2\langle a,b \rangle=\Vert a\Vert^2+\Vert b \Vert ^2 +2\Vert a \Vert \Vert b \Vert$$ $$\implies \langle a,b \rangle= \Vert a \Vert \Vert b \Vert.$$ And equality on Cauchy-Schwarz only holds if both vectors are linearly dependent. (Note that since there is no modulus on the inner product on the left side, not only they must be linearly dependent, but also differ by a positive scaling). As a sidenote, the technique above holds for any inner product space. One might be tempted to extend it to any normed space, but the result isn't true. As an example, one can take $L^1([0,1])$ and $a=I_{[0,1/2]}$, $b=I_{[1/2,1]}$, where $I_A$ is the indicator function on $A$. Sure; for example, in one dimension, a vector (+10) and a vector (-10) would have a sum of 0, and a difference of magnitude 10 - (-10) = 20. |
Can the magnitude of the sum of two vectors ever be greater than the magnitude of either of the vectors?
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