AP Calc BC: Vectors

AP Calculus Vectors

Definition

• A vector is a quantity that has both magnitude and direction.
• It is represented by an arrow with its tail at the origin and its head at the point representing the vector.
• Vectors can be added, subtracted, and multiplied by scalars.

Components of a Vector

• A vector in two-dimensional space has two components: x and y.
• A vector in three-dimensional space has three components: x, y, and z.

Vector Operations

• Addition: To add two vectors, add their corresponding components.
• Subtraction: To subtract two vectors, subtract their corresponding components.
• Scalar Multiplication: To multiply a vector by a scalar, multiply each component by the scalar.
• Dot Product: The dot product of two vectors is the product of their corresponding components added together.
• Cross Product: The cross product of two vectors is a vector that is perpendicular to both vectors and has a magnitude equal to the product of their magnitudes times the sine of the angle between them.

Applications

• Vectors are used in physics to represent forces, velocities, and accelerations.
• They are used in engineering to represent forces, moments, and torques.
• They are used in computer graphics to represent positions, directions, and velocities of objects.