Vectors Flashcards
How to show two vectors are perpendicular?
Show that dot product = 0 as the rest don’t matter snd cos 90 is 0
Cartesian equation of plane
Is ( r - a) dot n
So r .n - a.n
So n1x + n2 y +n3z + d where d = -a.n
Btw r is a general point in the plane, a is a position vector, such that r - a is the vector going from a to r
And this . With n must equal 0
So r-a dot n =0!
How to solve for d
I’d equation given in vector form, how to do
Sub point a in isntesd of doing d = -a . N
Find the n, remember it’s r -a dot n, so find n, that’s the n, and sub in a
Angle between two PLANES
Acute angle? + why is it giving an obtuse
= angle between normal vector!
2) if normal vecotr runs other way you’ll get obtuse angle given, each plane Has two obtuse and two actue
- just do 180- angle to get the ACTUTR angle between the planes
Equation if line vector?
Position vector (O TO A POINT)
Direction vector (point to other point ) and lambda
How to see if 3 points are collinesr?
Find ewaurikn of two points
Now see if third point matches this one
Check by substituting it for R, and here if it fits then all three eauatjons if lambda will be the SAME LAMBDA
How to do vector equation of line for Cartesian
It’s x - the positon / direction
Converting back, make sure it’s in this form. ALLOWED TO MULTIPLY. Y -1//-1 TO GET IT LIKE THAT!
Special cases for converting vector to Cartesian?
1 0= parameter = that position and others are related
2 0 means thst third parented can be anything, no = sign to put it next to so that’s how to remember to discard
Thus just do x = y = that’s it
Angle between plane AND LINE
Two steps!
plane you can see the normal vector, line you can see direction vector, and so the angle between normal and line is x and theta is 90-x!
But remember must have ACUTE ANGLE, so if the normal and line give you obtuse, 180- it, then 90- it!
DONT LACK
How to see if lines intersect in 3d?
If they intersect value of r which means xyz will be THE SAME
So equate them
- find equations for top middle bottom
- then simultaneously solve for mew and lands using first two equations
To check if they do intersect these values should work for the third equation
- To find point of intersection, must add these vakues if mew and lambda to the position vector and it should end up being the SAME POINT
Intersection line and plane?
R = xyz make three equations for xyz in terms of lambda and sub into the plane
Find lambda and put these back into xyz equations to find point
To check, sub point itno to plane. If it is a valid point on the plane it will equal 0!
How to know if two lines are parallel?
In 3d , if they aren’t parallel and don’t intersect what is it called
Parallel is it the direction vector will have some sort of multiple
If not parallel and don’t intersect they are SKEW
If a x b vector product is 0 what does that tell you and why
The only way for this to happen is if sin theta is 0, using alternative form of vector product
The intl way for sin theta to be 0 is if they are PARALLEL to each other!, thus if parallel there is no vector that is perpendicular to both and thus vector product is 0
What does vector proud t mean in terms of angle, x cross y?
Going from x to y ANTI CLOCKWISE
What does the vector product length even show
The length of the vector product is = to the area formed of parallelogram between vectors
So why is I cross j k ? Etc
What about j cross I
What does this show about vector product
I x j , the cross ProductCode will be equal to area, so 1 x 1 so 1, so K works as this is perpendicular to both
However j x I , you go 270 anti closokise so that becomed negative in sin, so it’s -k
As a result vector product is NOT COMMUTATIVE
When doing a x b how to write it done
How to prove they arentvomuutove
A x b means when doing determinant you write A FIRST
to prove not commutivity find det with a first, and then with b first and show they aren’t the same!
Rememebr Unit vector?
Is the vector / the modulus of the vector for each compoennt
How to find the area of a triangle using vector product then if origin
You know vecotr product length = area parallelogram
So area triangle = 1/2 area parallelogram =1/2 the mod of the VECTOR PRODUCT!
How to find equation of plane easy now using vecotr product
Need a vector perpendicular to play
Work out teo vectors (not OA ) and cross them
Then that’s your normal vector, and use point on equation to find d
Finally check third point equated to 0!
