Vectors are powerful mathematical tools that combine and direction. They're essential in physics, engineering, and computer graphics, allowing us to represent forces, velocities, and positions in space.
Understanding vectors is crucial for solving complex problems in multiple dimensions. We'll learn how to represent, manipulate, and analyze vectors, building a foundation for more advanced concepts in linear algebra and calculus.
Vector Fundamentals
Geometric and algebraic vector interpretation
Vectors are mathematical objects possessing both magnitude (length) and direction
Geometrically represented as directed line segments with an and a (arrow)
Algebraically represented as ordered pairs or triples (a,b) or (a,b,c) where a, b, and c are the 's components along the x, y, and z axes respectively
Vector magnitude and direction
Magnitude (length) of a vector v=(a,b) calculated using the formula ∣v∣=a2+b2
For a 3D vector v=(a,b,c), magnitude is ∣v∣=a2+b2+c2
Direction of a vector described by the angle it makes with the positive x-axis
Angle θ given by tanθ=ab where a and b are the vector's components (v=(a,b))
Vector addition and scalar multiplication
performed component-wise: v+w=(a+c,b+d) for v=(a,b) and w=(c,d)
In 3D space, v+w=(a+d,b+e,c+f) for v=(a,b,c) and w=(d,e,f)
is multiplying a vector by a (real number): kv=(ka,kb) for scalar k and v=(a,b)
In 3D space, kv=(ka,kb,kc) for v=(a,b,c)
Vector Representations and Operations
Vectors in component form
Vectors expressed in component form as v=(a,b) or v=(a,b,c)
a, b, and c represent the vector's components along the x, y, and z axes respectively
Component form simplifies algebraic manipulation and computation of vector operations (addition, subtraction, scalar multiplication)
Unit vectors in given directions
is a vector with a magnitude of 1 pointing in a specific direction
Unit vector in the direction of v=(a,b) given by v^=∣v∣v=(a2+b2a,a2+b2b)
For 3D vector v=(a,b,c), unit vector is v^=(a2+b2+c2a,a2+b2+c2b,a2+b2+c2c)
I and j notation for vectors
Standard unit vectors in 2D space denoted as i^=(1,0) and j^=(0,1)
In 3D space, standard unit vectors are i^=(1,0,0), j^=(0,1,0), and k^=(0,0,1)
Vector v=(a,b) written as v=ai^+bj^
In 3D space, v=(a,b,c) written as v=ai^+bj^+ck^
These standard unit vectors serve as basis vectors for the vector space
Dot product of vectors
of two vectors v=(a,b) and w=(c,d) given by v⋅w=ac+bd
For 3D vectors v=(a,b,c) and w=(d,e,f), dot product is v⋅w=ad+be+cf
Dot product results in a scalar value and determines the angle between two vectors
v⋅w=∣v∣∣w∣cosθ where θ is the angle between the vectors
If dot product is 0, vectors are perpendicular (orthogonal) to each other (90° angle)
Dot product can be used to calculate the projection of one vector onto another
Vector Spaces and Linear Combinations
Vector spaces
A vector space is a set of vectors that is closed under addition and scalar multiplication
Vector spaces can be described using linear combinations of basis vectors
A vector field is a function that assigns a vector to each point in a region of space
Linear combinations
A linear combination is the sum of scalar multiples of vectors
Any vector in a vector space can be expressed as a linear combination of its basis vectors
Key Terms to Review (14)
Dot product: The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is calculated as the sum of the products of corresponding entries.
Initial point: The initial point of a vector is the starting point from which the vector originates. It is often denoted as the coordinate $(x_1, y_1)$ in a two-dimensional plane or $(x_1, y_1, z_1)$ in three-dimensional space.
Magnitude: Magnitude represents the size or length of a vector. It is always a non-negative value and can be found using the Pythagorean theorem in two or three dimensions.
Parallelograms: A parallelogram is a quadrilateral with opposite sides that are both parallel and equal in length. Its opposite angles are also equal.
Position vector: A position vector is a vector that represents the position of a point in space relative to an origin. It is typically written in the form $\vec{r} = \langle x, y, z \rangle$ in three-dimensional space.
Resultant: The resultant is the vector that represents the sum of two or more vectors. It provides both the magnitude and direction of this combined effect.
Scalar: A scalar is a quantity that is fully described by a magnitude alone. It does not have any direction.
Scalar multiple: A scalar multiple is the product of a scalar (a real number) and a vector or matrix. It scales the magnitude of the vector or matrix without changing its direction.
Scalar multiplication: Scalar multiplication involves multiplying a vector by a scalar (a single number), resulting in a new vector where each component is scaled by the scalar. This operation changes the magnitude of the vector but not its direction if the scalar is positive.
Standard position: An angle is in standard position if its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The angle's terminal side then determines its measure.
Terminal point: The terminal point is the ending point of a vector, typically represented in coordinate form. It indicates the final position after moving from the initial point according to the vector's direction and magnitude.
Unit vector: A unit vector is a vector with a magnitude of 1. It is often used to indicate direction without considering the magnitude.
Vector: A vector is a quantity that has both magnitude and direction. It can be represented graphically as an arrow or algebraically as an ordered pair or triplet.
Vector addition: Vector addition is the process of combining two or more vectors to form a resultant vector. The resultant vector represents the cumulative effect of all vectors involved.