Vectors for Calculus-Based Physics AP Physics C

A Vector … … is a quantity that has a magnitude (size) AND a direction. …can be in one-dimension, two-dimensions, or even three-dimensions …can be represented using a magnitude and an angle measured from a specified reference …can also be represented using unit vectors

Vectors in Physics 1 We only used two dimensional vectors All vectors were in the x-y plane. All vectors were shown by stating a magnitude and a direction (angle from a reference point). Vectors could be resolved into x- & y-components using right triangle trigonometry (sin, cos, tan)

Unit Vectors A unit vector is a vector that has a magnitude of 1 unit Some unit vectors have been defined in standard directions. +x direction specified by unit vector “i” +y direction specified by “j” +z direction specified by “k” “n” specifies a vector normal to a surface

Using Unit Vectors For Example: the vector The hat shows that this is a unit vector, not a variable. is three dimensional, so it has components in the x, y, and z directions. The magnitudes of the components are as follows: x-component = +3, y-component = -5, and z-component = +8

Finding the Magnitude To find the magnitude for the vector in the previous example simply apply the distance formula…just like for 2-D vectors in Physics 1 Where: Ax = magnitude of the x-component, Ay = magnitude of the y-component, Az = magnitude of the z-component

Finding the Magnitude So for the example given the magnitude is: What about the direction? In Physics 1 we could represent the direction using a single angle measured from the +x axis…but that was only a 2D vector. Now we would need two angles, 1 from the +x axis and the other from the xy plane. This is not practical so we use the i, j, k, format to express an answer as a vector.

Vector Addition If you define vectors A and B as: Then:

Example of Vector Addition If you define vectors A and B as: Note: Answer is vector!

Vector Multiplication Dot Product Cross Product Also known as a scalar product. 2 vectors are multiplied together in such a manner as to give a scalar answer (magnitude only) Also known as a vector product. 2 vectors are multiplied together in such a manner as to give a vector answer (magnitude and direction)

Finding a Dot Product If you define vectors A and B as: Then: Where Ax and Bx are the x-components, Ay and By are the y-components, Az and Bz are the z-components. Then: Answer is a magnitude only, no i, j, k unit vectors.

Example of Dot Product If you define vectors A and B as: Note: Answer is magnitude only!

Dot Products (another way) If you are given the original vectors using magnitudes and the angle between them you may calculate magnitude by another (simpler) method. A Where A & B are the magnitudes of the corresponding vectors and θ is the angle between them. θ B

Using a Dot Product in Physics Remember in Physics 1…To calculate “Work” Where F is force, d is displacement, and  is the angle between the two. Now with calculus: Dot product of 2 vector quantities Note: This symbol means “anti-derivative”… we will learn this soon!

Finding a Cross Product If you define vectors A and B as: Where Ax and Bx are the x-components, Ay and By are the y-components, Az and Bz are the z-components. Then: Answer will be in vector (i, j, k) format. Evaluate determinant for answer!

Example of a Cross Product If you define vectors A and B as: Set up the determinant as follows, then evaluate.

Evaluating the Determinant One way to evaluate this determinant is to copy the first 2 columns to the right of the matrix, then multiply along the diagonals. The products of all diagonals that slope downward left to right are added together and products of diagonals that slope downward from right to left are subtracted. - - - + + Final answer in vector form.

Cross Products (another way) If you are given the original vectors using magnitudes and the angle between them you may calculate magnitude by another (simpler) method. Note: the direction of the answer vector will always be perpendicular to the plane of the 2 original vectors. It can be found using a right-hand rule! A Where A & B are the magnitudes of the corresponding vectors and θ is the angle between them. θ B

Using a Cross Product in Physics Remember in Physics 1…To calculate “Torque” Where F is force, l is lever-arm, and  is the angle between the two. Now with calculus: Cross product of 2 vector quantities r is the position vector for the application point of the force measured to the pivot point.

Some interesting facts The commutative property applies to dot products but not to cross products. Doing a cross product in reverse order will give the same magnitude but the opposite direction!