Math 151- Vectors, Projections, Inverse Trig Key Formulas

What is the domain and range of arcsin(theta)?

D : [-1,1]

R : [-pi/2, pi/2]

What is the domain and range of arccos(theta)?

D : [-1,1]

R : [0,pi]

What is the domain and range of arctan(theta)?

D : (negative infinity, positive infinity)

R : (-pi/2, pi/2)

What is the formula for finding the length ( magnitude ) of a a vector?

sqrt(x² + y²) where x and y are the components of the vector.

What is the formula for Resultant Force/ Resultant Vector ( adding vectors )?

(component a1 + component a2), Component b1 + component b2 )

Note: the answer would create 1 x component and 1 y component that would be in the vector parenthesis.

What is the formula for multiplying a vector by a number.

The constant multiplied by the x and y component of the original vector, leaving a new vecotor.

What is the formula for finding perpendicular ( orthogonal) vectors.

when given vector x,y you can do -y,x to get a perpendicular vector.

A more useful method is doing (vector a * vector b) = 0.

What is the formula for the Dot Product ( multiplying vectors ) ( there is 2)

When given components: (component x1 times component x2) + (component y1 times component y2)

when given angle: mag(a) mag(b) * cos(theta)

How do you find the x and y component of a vector when given an angle?

mag(v) * cos(theta), mag(v) * sin(theta)

Note: this is a vector.

How do you find the angle between 2 vectors?

cos(theta) = dot product / magnitudes of vector a and vector b multiplied by each other.

Will need to do arccos to find angle.

What is the formula for finding a unit vector and why is it useful?

The unit vector allows us to create whatever length we want and to create a vector from a line by getting direction without changing lengths.

vector a / mag(a)

What is the formula for work done?

Both versions of the dot product are used for work done, however, the variables are different.

F = force( vector a)

D = Displacement ( vector b )

What is the formula for scalar projections?

(dot product of a and b) / (mag(a)) when b is projected onto a.

This equation is also known as length when talking about projections, the answer will generate a value.

What is the formula for Vector Projections?

( dot product of a and b ) / (mag(a)) all times the unit vector of vector a when b is projected onto a.

This equation is the scalar projection being multiplied by a unit vector so that the answer is a vector instead of a value.