math 2d midterm 1

how do you add 2 vectors

add component by component, u + v = <u1+v1, u2+v2, u3+v3>

how do you subtract 2 vectors

subtract component by component, u - v = <u1-v1, u2-v2, u3-v3> (vector that’s the minuend is the number that is subtracted from)

what does this symbol mean: | |

magnitude (aka length)

how do you calculate |u| (aka how do you find the magnitude of a vector)

take the square root of the sum of the squared components to find the size of vector u, meaning u=<a,b,c> therefore |u| = √(a²+b²+c²)

what does this mean: |u+v|

add the vectors u and v together to get one combined vector, and measure their final length

what does this mean: |u| + |v|

find the length of u, find the length of v, and add them together.

triangle inequality

the sum of the lengths of any two sides of a triangle must be greater than or equal to the lengths of the remaining side

when do vectors point in the same direction? what does this indicate?

two vectors point in the same direction if one is a positive scalar multiple of the other (v=cu). if they point in the same direction, they are parallel and the magnitude of their sum is equal to the sum of their magnitudes (they can also be parallel positively and negatively).

what’s the difference between saying two vectors are the same direction vs two vectors are the opposite direction

if two vectors are the same direction, one vector is multiplied by a positive scalar multiple. whereas if two vectors are in opposite directions, it means a vector is multiplied by a negative scalar.

parallelogram law (explanation)

two vectors u and v are drawn from the same starting point, forming two adjacent sides of a parallelogram where the length of one diagonal (referred to as the main diagonal) is u+v and the length of another (referred to as the cross diagonal) is u-v. the sum of the squares of the lengths of the two diagonals = the sum of the squares of the lengths of all four sides

parallelogram law (write the equation)

|u+v|²+|u-v|²=2|u|²+2|v|²

how do you determine if two non-zero vectors u and v are orthogonal (perpendicular)

u•v=0, their dot product must be 0

how do you determine if two non-zero vectors u and v are parallel

if one vector is a scaled version of the other vector, or if their cross product is 0 (u x v = 0)

how do you multiply a vector by a scalar (and consider negative scalars)

multiply each component by the scalar; a negative scalar (c<0) switches the direction of the vector

how do you solve |cu| (taking the magnitude of a vector with a scalar multiple)

|c| * |u| where |c| is the absolute value of c (the scalar)

how to take the dot product of two vectors

u*v = u1*v1 + u2*v2 + u3*v3

how do you find the angle between 2 vectors u and v (or 2 planes n1 and n2)

geometric definition of the dot product; cos(x) = (u*v)/(|u||v|) and solve by isolating x using arccos

how do you calculate |u|²

u*u (self dot product)

how to take cross product of two vectors

create a 3×3 matrix, diagonal = ad-bc, form i-j+k

how do you find the area of a parallelogram

cross product; |u x v|

how do you find the area of a triangle

cross product; ½ |u x v|

how do you find the normal vector given a plane’s equation

the coefficients of x, y, and z are the components of the normal vector

how do you find the normal vector given 3 points on a plane

create two vectors using the 3 points and find the cross product of the two vectors

when are two lines parallel

when the cross products of their direction vectors are 0

when are two lines perpendicular

when the dot product of their direction vectors is 0

when is a line parallel to a plane

when the dot product of the line’s direction vector and the plane’s normal vector is 0

when is a line perpendicular to a plane

when the cross product of the line’s direction vector and the plane’s normal vector is 0

if 2 planes are parallel to a line, do they have to be parallel to each other

no

is dot product commutative

yes

is cross product commutative

no, it needs a neg sign (u x v = -v x u)

if a•b = a•c, does this mean b=c?

no, b doesn’t have to equal c because you cannot use the cancellation law on vectors (you can’t divide vectors) where you divide both sides by a to come to the conclusion that b=c. if a happens to be a vector that is perpendicular to b and perpendicular to c, they would have the same dot product of 0 but they wouldn’t be the same vector. a could also be some vector with majority 0s so you’d only be looking at one component of the vector because he other 2 cancel out (and b and c could have the same x or y or z component while not being the same vector).

what constraints are put on a vector to be a straight line in 3d space

the direction vector cannot change its proportional directino as t changes, meaning each component needs to be a linear function of t or have the exact same scalar multiple (ie all 3t³) for each component. if they have scalar multiples that are different like t raised to different powers or different trig functions (basically if you cannot factor out the t to leave behind constant components), it cannot be a line.

what is a level surface of a 3d function

setting a function f equal to some constant k, f(x,y,z)=k

are two planes parallel if they have the same normal vector

yes because that means they point the same direction in 3d space and their cross product is 0, since two vectors are parallel if one is a scalar multiple of the other (so if they’re the same, they’re scaled by the scalar multiple 1)

are the magnitudes of the following cross produces equal?
|u x v| = |v x u|

yes, because u x v = -v x u, but in magnitude length can never be neg so |u x v| = |-v x u| becomes |u x v| = |v x u|

what is the equation for the second derivative of a parametric equation, given that your x and y equations are given in terms of a third parameter such as t

d²y/dx² = d/dt(dy/dx)/(dx/dt) (derive dy/dx again and divide that by the first derivative of x)

how do you find points where the curve has a horizontal tangent line/curve is || to x-axis

find the value of t where the numerator dy/dt is 0, given that the denominator dx/dt isn’t 0 (because then it would be 0/0, undefined) and plug t into the original equation

how do you find points where the curve has a vertical tangent line/curve is || to y-axis (or perpendicular to x-axis)

find the value of t where dx/dt=0, given that dy/dt isn’t 0 (because then it would be 0/0, undefined) and plug t into the original equation

what can u tell from the second derivative

if its positive, ur graph is concave up, and if its negative, ur graph is concave down

scalar triple product

used to calculate the volume of 3d shapes, most notably a parallelepiped (|a•(b x c)|). the scalar triple product can also be solved as a determinant using cofactor expansion since it’s a 3×3 matrix

how do you prove 3 vectors are coplanar (lie on the same plane)

the scalar triple product is 0, because if they were on the same plane that means we are working with a 2d shape and 2d shapes in 3d have a volume of 0

how do you check if two lines in 3d intersect or are skew

create a 2×2 system by setting the components of each vector equal to each other, and use two equations to solve for your parameters (usually t and s). then plug your parameter values into the unused equation, if they are equal they intersect and if not they are skew.

how to check if 2 lines intersect at a point given r(t), the equation of the line (remember this usually consists of the initial point and the direction vector which has a 3rd parameter)

create a 2×2 system of eq, solve for the parameters, and plug any parameter into its corresponding direction vector (doesn’t matter which one) to get coordinate points

how to check if 2 lines are the same line

check whether the direction vectors are scalar multiples, and create a series of equations by setting the initial point of the first line equal to each component of the second line. if you get the same values for all 3 parameters, then your lines are the same (and if not they’re distinct)

what is the equation of a plane

a(x-x0) + b(y-y0) + c(z-z0), find the normal vector for your <a,b,c> values and plug in a given point for your x,y, and z coordinates

given 3 points, how do you find the equation of a plane

build 2 vectors using the 3 points (like ab and ac) and subtract their components, find the normal vector using the cross product, and plug corresponding values into the plane equation

how to find the parametric line of intersection for 2 planes

find the cross product of the 2 plane equations, pick a variable x,y, or z and set it =0, plugging it into both the plane equations and using a system of eq to solve for the other two variables. plug the point and direction vector into x=xo+at, y=y0+bt, and z=0+ct

how do you find the point on the curve where the tangent line is parallel to the plane

calculate the first derivative of the plane equation, and find the dot product of that (the tangent line) and the normal vector (which are just the components of the plane eq). then plug your t value back into the original position vector