### Precalculus – Vectors

A quantity with both magnitude and direction.

Vector

v = < a, b >

Where a and b are components of vector v.

Where a and b are components of vector v.

Algebraic Vector

One of two parts of a vector. In vector v = < a, b >, a and b are components, or coordinates, of vector v.

Component

The first component of a vector, representing the horizontal aspect, or x-coordinate, of a vector. Distance along the x-axis for a position vector.

Horizontal Component

The second component of a vector, representing the vertical aspect, or y-coordinate, of a vector. Distance along the y-axis for a position vector.

Vertical Component

v = ai + bj

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

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

Scientific Vector

The length of a vector, represented as ‖v‖. For vector v = < a, b >:

‖v‖ = √(a² + b²)

‖v‖ = √(a² + b²)

Magnitude

A vector u for which ‖u‖ = 1 is called a unit vector.

Unit Vector

The vector v whose magnitude is 0 and is assigned no direction.

Zero Vector

The segment of a line defined by two points.

Line Segment

A line segment bounded by points P and Q, so that the points proceed from P to Q, or vice versa. Definition of a Geometric Vector.

Directed Line Segment

A vector whose initial point is on the origin. For vector v = < a, b >, the initial point is at (0,0) and the terminal point is at (a,b).

Position Vector

A vector representing the direction and speed of an object.

Velocity Vector

A vector representing the direction and amount of force acting on an object.

Force Vector

The angle α between position vector v and the x-axis. 0° ≤ α ≤ 360°.

Direction Angle

v = ‖v‖(cos αi + sin αj)

Where v is a vector, ‖v‖ is the vector’s magnitude, and α is the vector’s direction angle.

Where v is a vector, ‖v‖ is the vector’s magnitude, and α is the vector’s direction angle.

Finding a Vector from its Direction and Magnitude

With relationship to vectors, scalars are any real numbers; quantities that have only magnitude.

Scalars

The result of multiplying a scalar by a vector. For scalar α abd vector v, scalar multiple αv is defined as:

1. If α > 0, αv is the vector whose magnitude is α times the magnitude of v and whose direction is the same as v.

2. If α < 0, αv is the vector whose magnitude is |α| times the magnitude of v and whose direction is opposite that of v. 3. If α = 0 or if v = 0, then αv = 0.

1. If α > 0, αv is the vector whose magnitude is α times the magnitude of v and whose direction is the same as v.

2. If α < 0, αv is the vector whose magnitude is |α| times the magnitude of v and whose direction is opposite that of v. 3. If α = 0 or if v = 0, then αv = 0.

Scalar Multiple

Let v = a₁i + b₁j:

αv = (αa₁)i + (αb₁)j = <αa₁, ab₁>

αv = (αa₁)i + (αb₁)j = <αa₁, ab₁>

Scalar Multiplication of Vectors

For any nonzero vector v, the vector,

u = v / ‖v‖

is a unit vector that has the same direction as v.

u = v / ‖v‖

is a unit vector that has the same direction as v.

Unit Vector in the Direction of v

The multiplication of two vectors that returns a scalar as the output. Let v = a₁i + b₁j and w = a₂i + b₂j:

v * w = a₁a₂ + b₁b₂

v * w = a₁a₂ + b₁b₂

Dot Product

If u and v are two nonzero vectors, the angle θ, 0 ≤ θ ≤ π, between u and v is determined by the formula

cos θ = (u * v)/(‖u‖ ‖v‖)

cos θ = (u * v)/(‖u‖ ‖v‖)

Finding the Angle between Vectors

Describes two vectors which meet at a right angle, synonymous with “perpendicular” and “normal.”

Orthogonal

Two vectors v and w are orthogonal if and only if

v * w = 0

v * w = 0

Determining Whether Vectors are Orthogonal

Let vector v and vector w be two nonzero vectors with the same initial point. To decompose vector v is to derive two vectors v₁ and v₂ from vector v with respect to vector w, which are parallel and orthogonal to vector w, respectively.

v₁ = ((v * w)/(‖w‖²))w

v₂ = v – v₁

v₁ = ((v * w)/(‖w‖²))w

v₂ = v – v₁

Decomposition

When decomposing vector v, the vector v₁ which is parallel to w is referred to the vector projection of v onto w.

Vector Projection

In physics, the work W done by a constant force F is moving an object from point A to a point B is defined as

W = (magnitude of force)(distance) = (‖F‖)(‖→AB‖)

It is assumed that the force F is applied along the line of motion. Otherwise, if the work done is at an angle θ to the line of motion, the work W done by F is defined as

W = F * →AB

W = (magnitude of force)(distance) = (‖F‖)(‖→AB‖)

It is assumed that the force F is applied along the line of motion. Otherwise, if the work done is at an angle θ to the line of motion, the work W done by F is defined as

W = F * →AB

Work