Math VECTORS

What are the two equations involving area of triangles?

(magnitude of vector a)(magnitude of vector b)sinx=magnitude of the cross product of a and b A=1/2(magnitude of cross product of a and b

What are the possible relationship between lines and how are they shown through matrices?

-Intersecting: 1 solution, at least two non zeros in each row -Parallel: no solution, all but one non zero in a row -Skew: no solution -Coincident: infinite/many solutions, row of zeros

how do you find angle between lines

-find direction for each (in vector form or parametric) -cosx equation using both directions

how do you find area of parallelogram?

-find last coordinate by setting one vector equal to another including that last coordinate -split parallelogram into two triangles -use Area equation

how do you find new velocity vector when given a speed?

-find magnitude of original velocity vector to find speed -for what scalar do you have to multiply old speed to get new speed -use that scalar and multiply x y (and z) to original velocity vector

Given vector equations, how do you find coordinates?

-take two vector equations with a common coordinate, change to parametric, and set equal to each other (x and y's) -elimination to solve for one parameter -plug parameter back into one of the original vector equations that has that parameter -result is coordinate

how do you find the shortest distance from a point to a line?

-write general coordinates for foot: basically x and y and z parametric of the original position vector -dot product of direction of object and direction of distance all equal to 0 *direction of object: is the velocity of object (from original position vector) *direction of distance: take general Foot coordinate and subtract the given point (perpendicular to Foot which is the shortest point) -solve for t and plug back into direction of distance -find magnitude of direction of distance to find shortest distance

Given two lines, classify them and find acute angle between them.

1) change both equations into parametric 2) write directions of both to see if they are equal (if they're off by a common scalar they are equal) -same direction: parallel or coincident -different direction: intersecting or skew 3) set all x y and z equal to each other 4) pick two, and solve for parameter 5) plug back into one of the ones just used to find other parameter 6) plug both parameters into original parametric 7) same direction: -if there is no solution when solving for parameters, they are parallel -if there is a solution for parameters, they are coincident 8) different directions: -if x y and z are equal they intersect at the point -if they are not equal, they are skewed

How do you find N (foot) given point and equation of plane?

1) find normal vector by taking cross product of both directions in plane equation 2) plug point (from plane equation) into ax+by+cz=d to find d 3) write equation of normal line using given point and direction (normal vector) 4) change to parametric and plug in x y and z into plane equation to find t 5) plug t back into parametric to find x y z which is N (foot)

How do you find when a line and plane intersect?

1) write line eqn going through plane using given point and the direction is the normal vector 2) write in parametric 3) substitute x y and z in parametric into plan equation 4) solve for t 5) plug t back into parametric to find intersection point (x, y, z)

What does it mean when it asks for "what form do infinitely many solutions have?"

equations in parametric for each variable (x y and z). usually let z=t and substitute for others.

how do you find an object's speed?

magnitude of velocity (direction)

How do you find angle between 2 planes?

same as line but absolute value

How do you find angle between line and plane?

sinx=|(dot product of direction of line and normal vector)/(multiplied magnitudes)|

When are A B and C collinear?

vector AB=k(vector BC)

How do you write equation of a plane?

vector: <x y z>= <point> + <direction>t + <non parallel direction> s cartesian: ax+by+cz=d <a b c> is the normal vector to the plane -find this by taking the cross product of two vectors on the plane (in other words the cross product of the direction vectors in vector form) -plug in initial x y and z to find d value

How do you write a line in vector, parametric, and cartesian?

vector: <x y z> = <point> + <direction>t parametric: x=(x point) + (x direction)t (SAME FOR Y AND Z cartesian: solve parametric equations for t and set all equal to eachother