0
Can the sum of magnitudes of two vectors ever be equal to the magnitude of the sum of these two vectors?
Wiki User
∙ 11y ago
Sure, if the two vectors point in the same direction.
When we need the sum of magnitudes of two vectors we simply add the magnitudes, but to get the magnitude of the sum of these two vectors we need to add the vectors geometrically.
Formula to find magnitude of the sum of these two vectors is sqrt[ |A|2 +|B|2 +2*|A|*|B|*cos(z) ] where |A| and |B| are magnitudes of two A and B vectors, and z is the angle between the two vectors.
Clearly, magnitude of sum of two vectors is less than sum of magnitudes(|A| + |B|) for all cases except when cos(z)=1(for which it becomes = |A| + |B| ). Cos(z)=1 when z=0, i.e. the vectors are in the same direction(angle between them is 0).
Also if we consider addition of two null vectors then their sum is zero in both ways of addition.
So, we get two cases
- when the two vectors are in same direction, and
- when the two vectors are null vectors.
In all other cases sum of magnitudes is greater than magnitude of the sum of two vectors.
Wiki User
∙ 11y ago
Add your answer:
Earn +20 pts
Can the sum of the magnitudes of two vectors ever b equal to the the sum of these two vectors?
Not really. The sum of the magnitudes is a scalar, not a vector - so they can't be equal. But the sum of the two vectors can have the same magnitude, if both vectors point in the same direction.
Can the sum of the magnitudes of two vectors ever be equal to the magnitudes of the sum of these two vectors?
only if the vectors have the same direction
Suppose you have two vectors that have different magnitudes can the vectors sum ever be zero?
No. The largest possible resultant magnitude is the sum of the individual magnitudes.The smallest possible resultant magnitude is the difference of the individual magnitudes.
Can the sum of of the magnitudes of two vectors ever be equal to the magnitudes of the sum of these two vectors?
No, they could be equal If the two vectors are opposites (180 degrees apart) like r and -r, then the sum of their magnitudes is the magnitude of their sum. ?? North 1 plus East 1 gives NorthEast 1.414. North 1 plus South 1 gives 0. North 1 plus North 1 gives North 2, which is equal to, not less than 1+1.
Can a pair of vectors with unequal magnitudes ever add to zero?
No.
Can the magnitude of the difference between two vectors ever be greater than the magnitude of either vector?
No.
Can the sum of two or three vectors ever be zero?
Yes. Two vectors that have equal magnitude and point in opposite directions have a sum of zero. (Like <1,0> and <-1,0>, one pointing in the positive x direction and one in negative x direction. The same idea applies with three vectors. For example, <1,0,0>, <-1,1,0> and <0,-1,0> have a sum of <0,0,0>.
Can the magnitude of a vector be ever equal to one of its components?
Yes. - if all the other components are zero. When the word "component" means the mutually perpendicular vectors that add (through vector addition) to form the resultant, then then answer is that "the magnitude of a vector" can equal one of its components, if and only if all other components have zero length (magnitude). This answer applies to the typical case of a vector being expressed in terms of components defined by an orthogonal basis. In normal space, these basis vectors merely define the relevant orthogonal coordinate system. There are, however, mathematical systems that use a nonorthogonal basis and the answer is different and presumably not part of the submitted question.
Can the magnitude if the difference between two vectors ever be greater than the magnitude of their sum?
yes,if the components are making angle 0<=theta<=90 no ,the magnitude of vector can never attain a negative value |a|=square root of both components which always gives a positive value
What scale is used to measure the energy released by an earthquake?
The moment magnitude scale is used by seismologists to measure the amount of energy released by large earthquakes (those greater than magnitude 8.0). For smaller earthquakes (those with magnitudes less than 7.0 and with epicentres less than 650 km from a seismometer station may be used) the method devised by Richter (the Richter magnitude scale) may be used to estimate the magnitude. The surface wave magnitude scale may be used for earthquakes with magnitudes up to 8.0 (devised by Richter and Gutenberg to extend the utility of the Richter scale.) Richter magnitudes are generally easier to derive than moment magnitudes being based on direct seismometer measurements, whereas the moment magnitude is a more4 fundamental measurement of magnitude being based on the rock mass strength around the fault, the amplitude of fault movement and the cross sectional area of that portion of the fault that moved. However this is more difficult to measure. As such it is common for initial reports to be in Richter magnitudes and more detailed letter magnitudes to be reported as moment magnitudes.
What type of boundary has the most earthquakes with high magnitudes?
Convergent boundaries where large scale thrust faulting occurs tend to have the largest magnitude earthquakes. For wxample the subduction boundary between the Pacific plate and the South American plate was responsible for the largest magnitude earthquake ever recorded (the magnitude 9.5 Valdivia earthquake that occurred in 1960 in Chile).
Can the magnitude of a resultant vector ever be less than the magnitude of one of its components?
Yes, if the two vectors are at a sufficiently large obtuse angle.The law of cosines gives the size of the resultant.If C = A + B, where A, B, C are vectors, then C is the "resultant."The law of cosines says, he magnitudes, A,B,C, are related as follows,C2=A2+B2+2AB cosine(theta),where theta is the angle between the vectors A and B. When theta is zero, then C has the maximum length, equal to the lengths of A and B added. When theta is 180 degrees, then C has the minimum length of the difference of the length of A and of B. Somewhere in between, the length of C will equal the length of the longer component and for larger angles be smaller.To be specific, suppose that A is the longer of the two, then the resultant, C, has the same length as A at one special angle which we will call theta*.A2=A2+B2+2AB cosine(theta*)cosine(theta*)=-B/(2A).The answer to the question is then, that for angles greater than theta* the resultant is smaller than the larger component. (Greater means, of course, greater than theta* and up to 360-theta*.)Note that if we ask whether the resultant can be smaller than the smaller of the two component vectors, then the answer is again yes and the above equation holds true when A is the smaller with the condition that it is not smaller than half the length of B. When the smaller vector is less than half the length of the larger component, then the resultant may equal the length of the larger but can never be made equal to the length of the smaller component.
