Math Formulas (Vectors)

To find the projection of U unto v

(U dot v/(|| v||)^2)v Ex: u= <1,-2> v=<3,1> ||v||=sqrt10 =(1/10)<3,1> = <3/10,1/10>

To find a vector with a given magnitude

First: Find Magnitude of given vector Next: Find Unit Vector Finally: Multiply Unit Vector by given magnitude EX: Find a vector in the same direction as V with a magnitude of 3 v = <4,3> ||V||= 5 Unit Vector = <4/5,3/5> <4/5,3/5>3 = <15/5,9/5> =<3,9/5>

When taking the derivative of an expression including an angle/ degree raised to a power

Leave the expression as written, THEN take chain rule: Ex: v= cot^2(t^3) v' =2 cot(t^3)*-csc^2(t^3)(3t^2) or v= cot(t^3) v' =-csc^2(t^3)(3t^2)

To find the angle between two angles

SET CALC TO DEGREES Take the inverse cosine of u dot v/magnitude of u x magnitude of v Ex: w= <5,0> v= <4,3> Angle = cos^-1 (20/(5)(5)) Angle = cos^-1 (4/5)

To find the direction angle

SET CALC TO DEGREES Take the inverse tangent of (y/x)

To find a vector in the opposite direction

Take the negative of the original vector EX: V= <4,2> Opposite Direction <-4,-2>

To find the magnitude

Take the square root of <x^2+y^2>

To find the component form of a vector when given the terminal and initial points:

Terminal-Initial

The speed of a particle equals

The magnitude of the vector

Orthogonal means

Two Perpendicular Vectors Occurs when the dot product of two vectors is equal to "0" *GRAPHING CALCULATOR PROBLEM* How to do graphing calculator portion: 2nd Trace Zero Left Bound Right Bound (Around point at which line crosses x-axis) Enter Value of "X" will appear

For bearing problems

V = <||v||cosu,||v||sinu> W= <||w||cosx, ||w||sinx> Use given weight/pounds/number as magnitude of v and u To find the resultant or magnitude of the two vectors: Add the vectors together To find the direction angle of the two vectors: Take the inverse tangent of (y/x) of the resultant vector

Equation of a quadratic

a(x-h)^2 +k

The derivative of sin is

cosx

The derivative of sec is

sec x tan x

The derivative of tan is

sec^2 x

Equation of a circle

(x-h)^2 + (y-k)^2 = r^2 Similar to Magnitude of Two Combined Vector: (v1+v2)^2 + (v2+u2)^2 = # On any circle with a radius of 10,

The derivative of csc is

-csc x cot x

The derivative of cot x

-cxc^2 x

The derivative of cos is

-sin x

To find the unit vector

Find the magnitude of the vector, and divide x and y by that magnitude. Check answer by finding the magnitude of the new vector EX: v = <3,4> ||v||= 5 Unit Vector: <3/5,4/5> Check answer: Find magnitude: sqrt (3/5)^2 + (4/5)^2 =1