How do you determine if a vector is parallel, perpendicular, or neither?

Is there like a one step process where you can determine it?





Take this for example:


u=%26lt;8,5%26gt;


v=%26lt;-2,4%26gt;





Is this vector parallel, perpendicular, or neither? What are the process for each? How do u know when its not parallel, perpendicular, or neither?|||Hi,





If it's perpendicular, its dot product or scalar product will equal zero.





For vectors %26lt;x1, y1%26gt; and %26lt;x2, y2%26gt;, the dot product is x1*x2 + y1*y2. For your vectors, it is 8(-2) + 5(4) = -16 + 20 = 4. They are not perpendicular.





If they were parallel there would be a scalar you could multiply times the x and y of one vector to equal the other vector.





-2 times -4 equals 8 for the x values, but 4 times -4 does not equal 5. Therefore they are not parallel either.





By process of elimination, they are neither.





I hope that helps!! :-)|||u have to finde the slope of it if it is paralell it has to be equal ,for perpendicular the multiple of them has to be -1|||If two vectors are perpendicular to each other, then the dot product is zero:





ux*vx + uy*vy = 0





In your example: 8*(-2) + 5*4 = -16 + 20 = 4, so these vectors are not perpendicular to each other.








magnitude of u: um = sqrt( ux^2 + uy^2 )


magnitude of v: vm = sqrt( vx^2 + vy^2 )





If two vectors are parallel, then the dot product is equal to the product of the magnitudes:


ux*vx + uy*vy = um * uv





In your example,


um = sqrt( 64 + 25 ) = sqrt ( 89) = 9.434


uv = sqrt( 4 + 16 ) = sqrt( 20 ) = 4.472


The product = 42.190, but the dot product is 4, therefore these vectors are not parallel.





The angle between the vectors can be found like this:





cos(Th) = dot product / (product of magnitudes)





For this example: cos(Th) = 4 / 42.190 = 0.0948


Th = 1.476 radians = 84.560 degrees.|||Well you can take (8,5) and (2,4) and just put them on graph paper, we just started this unit in school too|||It's been decades since I did geometry, but I think the direction of u is atan(8/5); the direction of v is atan(-2/4). So the angle between them is 84.5 deg. Does my brain still work after all that time?|||For a pair of vectors with non-zero components, simply divide the components in a like manner. If the values are identical, then the vectors are parallel. If the values are equal in absolute value, but opposite in sign, then they're perpendicular. Failing those two tests, you get "neither."





If you end up having to divide by zero (which you won't do), then you have infinite slope, so only if both had a zero in the corresponding components would the vectors be parallel; if they have zeroes in the opposite components, then they're perpendicular. Failing those two tests, you get "neither."