Precalc +: Unit Circle and Vectors

Unit Circle

Circle with a radius of 1

Terminal Point definitions on a Unit Circle

y = sin(θ)

x = cos(θ)

Terminal point: (cos(θ), sin(θ))

How to find points on the terminal ray

* Find the point on the unit circle
* Multiply coordinates by different values

How to justify a point being on the unit circle

Determine the radius. If it is 1, then the point is on the unit circle. If not, it is not.

How to determine a point on the unit circle on the terminal ray of θ given a point

* Find radius
* Divide each point by the radius to get r=1

Vector

Quantity that involves both magnitude (size) and direction. They are directed line segments, have an initial point (sometimes called the head) and a terminal point (sometimes called the head or tip,) and are usually denoted by **u** or a u with a line over it. They model motion and force where size/speed and direction both matter. The location of it doesn’t matter. If the length and direction are the same, then they are equal. There are an infinite number of starting points.

Scalar

Real # that measures magnitude, NOT direction.

Vector magnitude

||u|| Length of a vector. Calculated using the distance formula. From standard form, it is √(v₁² + v₂²).

Standard position of a vector

Vector whose tail is at the origin

Component Form of a Vector

Way of writing a vector through including both the horizontal and vertical components. In a vector with initial point P = (p₁, p₂) and terminal point Q = (q₁, q₂), the component form is <q₁-p₁, q₂-p₂>

Zero Vector

Vector whose magnitude is zero (has the same initial and terminal point)

Unit vector

Vector whose magnitude is one

Equivalent Vectors

Have the same components and the same direction and magnitude. They don’t have to have the same intitial and terminal points, but will be the same vector when graphed in standard form.

Scalar multiplication

Makes vectors longer or shorter. Scalars are distributed among all elements of a vector in component form.

Resultant Vector

Sum of two vectors. Corresponding components of two vectors are added. Can be visualized with the parallelogram or tip-to-tail method.

How to use the parallelogram method

* Draw the vectors so that their initial points coincide.
* Draw lines to form a complete parallelogram
* The diagonal from the initial point to the opposite vertex of the parallelogram is the resultant.

How to use the tip-to-tail method

* Draw the vectors one after another, placing the initial point of each successive vector at the terminal poiont of the previous vector.
* The vector from the initial point of the first vector to the terminal of the second vector is the resultant

Vector Subtraction

Subtract the corresponding components of each vector

How to find the unit vector given a vector

Divide the given vector by its magnitude

Standard unit vectors

Vectors i (horizontal, <1,0>) and j (vertical, <0,1>) Other vectors can be written as multiples of i or j.

Direction angle

The angle θ is the direction of a vector in standard position from the positive x-axis to the vector.

Reference angle of a direction angle

θ = tan^-1(v₂/v₁)

Represenation of the unit vector using sine and cosine

u = cos (θ)i + sin (θ)j

How to calculate any vector given magnitude and direction angle

v = ||v||(cos (θ), sin (θ))

Velocity vector

Includes magnitude and direction

Distance vector

(Velocity vector)(time/scalar)

Heading

The direction a body is pointing

Bearing

The actual direction a body is moving

Distance between two vectors

||u-v||

Dot product of two vectors

u • v = u₁v₁ + u₂v₂

Angle between two vectors

θ = cos^-1 \[(u • v)/(||u||||v||)\]

Orthogonal (perpendicular) vectors

If u and v are non-zero vectors and u • v = 0, then u and v are orthogonal

Parallel Vectors

If θ is the angle between u and v and cos (θ) = 1 (θ = 0) or cos (θ) = -1 (θ = 180,) then u and v are parallel