Vectors

<x-x1, y-y1, z-z1> dot <A, B, C> = A(x-x1) + B(y-y1) + C(z-z1) = 0

If vector P1P equals <x-x1, y-y1, z-z1>, then vector P1P times n equals...

no

Is cross product multiplication associative?

(a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k

a cross b where... a = a1i + a2j + a3k and b = b1i + b2j + b3k

quadrant bearing

an angle between 0 degrees and 90 degrees that is given as east or west of the north-south line (35 degrees east of south or S35°E)

<x2-x1, y2-y1>

component form of vector with initial point p1 and terminal point p2

principal unit vectors

i = <1,0> and j = <0,1>

component form

indicated by angle brackets; two numbers respectively represent how far the vector reaches in the x-direction and how far it reaches in the y-direction

initial point

the starting point of a vector

resultant vector

the sum of two or more vectors

parallel

two vectors are said to be this if there is a nonzero scalar that when multiplied by one equals the other vector; same or opposite direction; angle between is 0 degrees or 180 degrees

vector over the magnitude of the vector

unit vector formula

sum of two vectors

A + B : initial point of vector B at terminal point of vector A or initial points coincide and diagonal of parallelogram with same initial point is the resultant

yes

Do the vector distributive laws apply to cross product multiplication?

yes

Does the scalar distributive law apply to cross product multiplication?

vector

a quantity that has both magnitude and direction

true bearing

an angle measure given from true north and is given with three digits

torque

cross product can be used to find this vector quantity that measures how effectively a force applied to a lever causes rotation along the axis of rotation

parallel vectors

cross product of zero

magnitude-direction form / polar form

indicated by square brackets; explicitly lists magnitude and direction; magnitude is always non-negative and direction is measured from the perspective of 0 degrees representing due east and values increasing in counter-clockwise direction (unless otherwise stated)

-commutative -distributive -dot product of the vector and itself is the magnitude squared -the zero vector dot the vector is zero

properties of the dot product

reverse the direction / multiply by negative one

reversing the order of the factors in a cross product would do this to the resulting vector

dot product

used to compute scalar product of two vectors; multiply each component of one vector by the corresponding component of the other vector and add these products

cosine theta equals dot product over the product of the magnitudes

angle between two vectors

magnitude of the cross product of v and w

area of a parallelogram with adjacent sides vector v and vector w

work equals force vector dot distance vector which equals the magnitudes times cosine theta

equation for work done by a constant force in moving an object from point A to a point B

T = r x F

formula for torque

the square root of the sum of each component squared

magnitude formula

product of a vector and a scalar

magnitude of the absolute value of the scalar times the original vector; same or opposite direction depending on whether it is positive or negative

octants

planes divide the three dimensional space into eight regions, called these

magnitude of vector times cosine theta

projecting a vector onto the x-axis

magnitude of vector times sine theta

projecting a vector onto the y-axis

dot product over the magnitude of v squared, times v

projection of vector u onto vector v

vector subtraction

same as adding the opposite vector

equivalent vectors

same magnitude and direction

opposite vectors

same magnitude but opposite direction, always parallel (scalar multiple of negative one)

parallel vectors

same or opposite direction but not necessarily the same magnitude

terminal point

the ending point of a vector

magnitude

the length of a vector

first octant

the only named octant is the entirely positive octant, called this

zero vector

the resultant vector of adding a vector and its opposite

cross product / vector product

used to find the angle between vectors in space; result is a vector

direction

usually the angle the vector makes with a horizontal line

unit vector

vector with a magnitude of one

linear combination form

vector written using the principal unit vectors; combining horizontal and vertical components

absolute value of a dot b cross c

volume of a parallelepiped

orthogonal / perpendicular

when the dot product is zero two vectors are this; angle between is ninety degrees

standard position

when the initial point of a vector is at the origin