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Can the magnitude of a resultant vector ever be less than the magnitude of one of its components?
Wiki User
∙ 13y ago
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.
Wiki User
∙ 13y ago
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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.
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 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.
In maths C is a second order dominance matrix the same as a second order ranking?
No. To find the second-order ranking (second-order dominance vector, V2), you add each of the columns and it will give you a (how ever many rows) x 1 matrix.
Why do two magnetic lines never intersect each other?
It is important to realize that magnetic lines do not really exist! They are a tool to visualize the magnetic field, but the field is continuous and does not exist solely inside lines. The direction of the lines gives the direction of the magnetic field, the density of lines, its strength. This also explains why no two field lines can ever intersect; a field line carries information about the direction of the magnetic field, if they would intersect an ambiguity would arise about the direction (not to mention a field of apparent infinite strength since the density would be infinite at the point of crossing). The field lines are almost never used in explicit calculations; instead one uses a vector, an entity which contains information about the magnitude and direction of a field in every point in space and time. Adding two magnetic fields is then easy; just add the vectors of both fields in every point in space (and time). You can use the resulting vector field to draw field lines again if you want. An easy way to imagine what would happen to field lines when they might intersect is to look at them as being such vectors. Imagine you have one field line pointing to the right, and another one pointing up. The result of adding would be a field line pointing somewhere in the up-right direction (the exact direction depending on the relative magnitudes of the fields). If the fields are equal in magnitude but opposite in direction they would cancel; the field line disappears. But this is to be expected! The magnetic fields canceled each other in that point! One has to take care with this analogy however; as for field lines the measure of magnitude is their density; which is an undefined thing if you are considering just one field line per field. For a vector however, the measure of magnitude is its length. Therefore adding two field lines of the same magnitude and pointing in the same direction would result in a vector of twice the length, but in field line language you would have to double the density at that point. This is one of the reasons field lines are used for visualization but not calculation. By the way, all these things apply to other fields as well. Electric fields can also be represented by field lines, and they as well cannot intersect (for the same reasons). Electric field lines, however, are not necessarily closed loops like magnetic field lines (this has to do with the non-existence of magnetic monopoles).
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 component of a vector ever be greater than the magnitude of the vector?
No.
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
Can the magnitude of the difference between two vectors ever be greater than the magnitude of either vector?
No.
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 magnitudes of the difference between two vector ever be greater than magnitude of either vector?
Yes. For example, if one vector has a length of 1 and points north, and the second vector has a length of 1 and points south, the difference (vector1 minus vector2) will have a length of 2 and point north.
What was the magnitude of the strongest earthquake ever?
9.5
What earthquake magnitude is the highest ever?
The magnitude 9.5 earthquake that occurred in Valdivia Chile in 1960.
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.
What is the strongest magnitude earthquake ever measured on the Richter magnitude?
The strongest earthquake that can be measured using the Richter magnitude scale is one with a magnitude of 8.0. For earthquakes larger than this, the moment magnitude scale must be used.
When if ever can a vector quantity be added to a scaler quantity?
never
Can your speed and velocity ever have the same magnitude?
Speed and velocity always have the same magnitude, becausespeed is the magnitude of velocity.The difference is that velocity has a direction but speed doesn't
