Coordinate Systems and Vector Representation
A vector is an object that has both magnitude (length) and direction.
In a coordinate system, a vector can be represented as an ordered list of numbers, which represent
the projections of the vector onto each of the coordinate axes.
For example, in a 2D Cartesian coordinate system, a vector can be represented as (x, y), where x is
the projection onto the x-axis and y is the projection onto the y-axis.
Geometric Interpretation of Vector Operations
Vector addition is accomplished by adding the corresponding components of the vectors.
Scalar multiplication is accomplished by multiplying a vector by a scalar, resulting in a vector with the
same direction but a different magnitude.
Vector subtraction is accomplished by negating one of the vectors and then adding it to the other.
Dot product is a scalar that is the product of the magnitudes of the vectors and the cosine of the
angle between them.
Cross product is a vector that is perpendicular to both of the original vectors and has a magnitude
equal to the product of the magnitudes of the original vectors and the sine of the angle between
them.
Abstract View of Vectors and Linear Algebra
Vector space is a set of vectors that can be added and scaled, following certain rules.
Matrices can be used to represent linear transformations, such as rotations, translations, and scaling.
Linear transformations can be represented as a matrix multiplication with a vector.
Vector Addition and Scalar Multiplication
Vector addition is accomplished by adding the corresponding components of the vectors.
Scalar multiplication is accomplished by multiplying a vector by a scalar, resulting in a vector with the
same direction but a different magnitude.
Perspectives on Vectors
Vectors can be thought of as arrows in the plane or in space.
Vectors can also be thought of as displacements or directed line segments.
Vectors are used in physics to represent quantities such as velocity, acceleration, and force.
Vectors are also used in computer graphics to represent things such as position, direction, and color.
Additional information you might want to consider including in your notes:
A vector is an object that has both magnitude (length) and direction.
In a coordinate system, a vector can be represented as an ordered list of numbers, which represent
the projections of the vector onto each of the coordinate axes.
For example, in a 2D Cartesian coordinate system, a vector can be represented as (x, y), where x is
the projection onto the x-axis and y is the projection onto the y-axis.
Geometric Interpretation of Vector Operations
Vector addition is accomplished by adding the corresponding components of the vectors.
Scalar multiplication is accomplished by multiplying a vector by a scalar, resulting in a vector with the
same direction but a different magnitude.
Vector subtraction is accomplished by negating one of the vectors and then adding it to the other.
Dot product is a scalar that is the product of the magnitudes of the vectors and the cosine of the
angle between them.
Cross product is a vector that is perpendicular to both of the original vectors and has a magnitude
equal to the product of the magnitudes of the original vectors and the sine of the angle between
them.
Abstract View of Vectors and Linear Algebra
Vector space is a set of vectors that can be added and scaled, following certain rules.
Matrices can be used to represent linear transformations, such as rotations, translations, and scaling.
Linear transformations can be represented as a matrix multiplication with a vector.
Vector Addition and Scalar Multiplication
Vector addition is accomplished by adding the corresponding components of the vectors.
Scalar multiplication is accomplished by multiplying a vector by a scalar, resulting in a vector with the
same direction but a different magnitude.
Perspectives on Vectors
Vectors can be thought of as arrows in the plane or in space.
Vectors can also be thought of as displacements or directed line segments.
Vectors are used in physics to represent quantities such as velocity, acceleration, and force.
Vectors are also used in computer graphics to represent things such as position, direction, and color.
Additional information you might want to consider including in your notes:
