Precalculus – Vectors

A quantity with both magnitude and direction.
Vector
v = < a, b >
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

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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>
Scientific Vector
The length of a vector, represented as ‖v‖. For vector v = < a, b >:
‖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.
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.
Scalar Multiple
Let v = a₁i + b₁j and w = a₂i + b₂j:
v + w = (a₁ + a₂)i + (b₁ + b₂)j =
Addition of Vectors
Let v = a₁i + b₁j and w = a₂i + b₂j:
v – w = (a₁ – b₁)i + (a₂ – b₂)j =
Subtraction of Vectors
Let v = a₁i + b₁j:
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.
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
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‖)
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
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
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
Work

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