What is an anti-parallel vector?
To my knowledge, there is no such term as “an anti-parallel vector”. The usage is something as follows:Two vectors are said to be antiparallel, if they point in exactly opposite direction. The magnitudes of the two vectors NEED NOT BE THE SAME. What is important are the directions of the two vectors. If (l,m,n) are the direction cosines of a vector, its antiparallel vector will have its direction cosines as ( -l,-m,-n)
If two vectors, 2ī+3j-k and 4i+6j+xk are parallel to each other, what is the value of x?
Tiger Joe Sallmen and Larry Rothstein have already given the easiest way of determining the value of [math]x.[/math]There are two other ways (which are not as simple), as given below:Using the cross product.Since the vectors are parallel, their cross product is zero.[math]\Rightarrow \qquad (2\hat i + 3\hat j -\hat k) \times (4\hat i + 6\hat j + x\hat k) = 0.[/math][math]\Rightarrow \qquad \hat i(3x+6) - \hat j(2x+4) + \hat k(12-12) = 0.[/math][math]\Rightarrow \qquad 3x+6 = 0 \qquad[/math] and [math]\qquad 2x+4 = 0.[/math][math]\Rightarrow \qquad x = -2.[/math]Using the dot product.Since the vectors are parallel, the cosine of the angle between them is [math]1[/math].[math]\Rightarrow \qquad[/math] The dot product is equal to the product of the magnitudes of the two vectors.[math]\Rightarrow \qquad (2\hat i + 3\hat j -\hat k) \cdot (4\hat i + 6\hat j + x\hat k) = |2\hat i + 3\hat j -\hat k|\,\,|4\hat i + 6\hat j + x\hat k|.[/math][math]\Rightarrow \qquad 8+18-x = \sqrt{4+9+1}\sqrt{16+36+x^2}.[/math][math]\Rightarrow \qquad 26-x = \sqrt{14}\sqrt{52+x^2}.[/math][math]\Rightarrow \qquad (26-x)^2 = 14(52+x^2).[/math][math]\Rightarrow \qquad 676-52x+x^2 = 728+14x^2.[/math][math]\Rightarrow \qquad 13x^2+52x+52 = 0[/math][math]\Rightarrow \qquad 13(x+2)^2 = 0[/math][math]\Rightarrow \qquad x = -2.[/math]
Can negative vectors be parallel to each other?
There’s no such thing as a negative vector. In 2-dimensions, a vector can point in any of 360 degrees.Two vectors may be parallel, perpendicular, or askew, just like infinite lines. However, only vectors have direction.To answer your question, Two vectors may be parallel, but point in opposite directions. If the two vectors have the same magnitude, then the add to zero. In a sense one vector negates the other.
If two vectors are parallel to each other, what will be their cross product?
The angle between two parallel vector is zero. So the cross product of two parallel vectors is given by,A × B = |A| |B| sin (A,B) n,The cross product of two vectors A, and B is equal to the magnitude of vector A, multilied by magnitude of vector B, multiplied by sine of the angle between the vectors A and B, and n is a unit vector perpendicular to both A and B.Since the angle between the vectors A and B is zerio, the cross product of two vectors is zero vector perpendicular to the the two vectors. Since there is essentially one vector we cannot define a plane and take the direction of the vector obtained from cross product in accordance with right handed screw. The direction of the null vector obtained from two parallel vector is not unique.
How can I find unit vector parallel to the resultant?
A vector is made up 2 elements that are- its direction and its magnitude.2 vectors would be parallel if they point in the same direction. if it's pointing towards x axis, the parallel vector should also point towards x axis,ORif it's pointing towards the direction of 45 degrees from the x axis, the parallel vector should also point to the same direction.ANDa unit vector would be a vector, whose magnitude is 1.So to find a unit vector parallel to the given resultant vector, we just make its magnitude 1 and give it the same direction as described in the example below! Let's say that the resultant vector is 6i, which means the vector points towards only one direction. Now, a vector of unit magnitude, and parallel to this would clearly be i or 1i, because it has a magnitude of one and the direction is same as that of the vector mentioned above.We got the answer just by the formula you described yourselfR cap=6i(resultant)/|6|(magnitude)= iSame would go for even 2D or 3D vectors! Hope it helped!
When are two vectors at right angle to each other?
Say we have two vectors; A and B.We have a formula to compute angle between them. It’s given by the following;[math]Cos (θ) = (A. B) / ( ||A|| * ||B|| )[/math]where A . B can be obtained by finding the dot product of A and B and || . || is norm of the vector. So, ||A|| is norm (magnitude) of vector A and ||B|| is the norm (magnitude) of vector B.Now to get right angle (θ = 90°), we know [math]Cos 90° = 0.[/math]So, in this formula C[math]os (θ) = (A. B) / ( ||A|| * ||B||[/math] ), the only way you can get the right hand side to be 0 is if the numerator is 0.i.e. [math]A . B = 0[/math]So, vectors can have right angle between them if their dot product is 0.
What is the difference between colinear and parallel vectors?
Thanks for the A2A…Hope that helps.Peter James Thomas
Two vectors, A and B, are such that their cross product, A x B = 0. What does this imply about the two vectors?
Two vectors, A and B, are such that their cross product, A x B = 0. What does this imply about the two vectors?The answer says either A is parallel to B or A is perpendicular to B.How can A be perpendicular to B?They might have meant "anti-parallel" (i.e., along the same line but in the opposite direction). The only other way to have a zero cross product is if one of the magnitudes was zero to begin with. If at least one of the vectors is zero, that would technically also make the vectors perpendicular, maybe, depending on your definitions; the dot product would certainly be zero, which is one common test for whether two vectors are perpendicular. However, for non-zero vectors that are perpendicular to each other, the cross product will not be zero.To be concrete:[math]| \vec A \times \vec B| = |\vec A| \, |\vec B| \, | \sin (\theta) |[/math]where [math]\theta[/math] is the angle between the vectors. Therefore, a zero cross product implies that one (or more) of the three terms on the right is zero: either one of the original vector magnitudes, or the sine of the angle between them. In the latter case, this only works for 0° (parallel) or 180° (anti-parallel).
How magnetic fields work at the most fundamental level?
I think the new startling M-theory (aka SuperStrings theory) could solve that question. M-theory suggests that *everything*, matter and force, is made up of a single ingredient. An unimaginably small vibrating strands of energy called strings. In that context, the force is a string, so it can move out an reach other strings (matter) and affect them. The problem is that I don't know if these strings are a real-physical entities or just a mathematical model that helps us understand reality? If M-theory is right, and if these strings are real-physical entities, then the question would be easily solved.