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 Vector Notation An effective and popular system used in engineering is called unit vector notation. It is used to denote vectors with an x-y Cartesian coordinate system.

Unit Vectors A unit vector is a vector that has a magnitude of 1 unit Some unit vectors have been defined in standard directions. The proper terminology is to use the “hat” instead of the arrow. So we have i-hat, j-hat, and k-hat which are used to describe any type of motion in 3D space.

Unit Vector Notation =3j = 4i The hypotenuse in Physics is called the RESULTANT or VECTOR SUM. The LEGS of the triangle are called the COMPONENTS 3j Vertical Component 4i Horizontal Component

Unit Vector Notation How would you write vectors J and K in unit vector notation?

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) To be learned at a later time

Cross Products (the simple 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.

The “Cross” Product (Vector Multiplication) Multiplying 2 vectors sometimes gives you a VECTOR quantity which we call the VECTOR CROSS PRODUCT. In polar notation consider 2 vectors: A = |A| < θ1 & B = |B| < θ2 The cross product between A and B produces a VECTOR quantity. The magnitude of the vector product is defined as: Where q is the NET angle between the two vectors. As shown in the figure. q A B

The significance of the cross product In this figure, vector A has been split into 2 components, one PARALLEL to vector B and one PERPENDICULAR to vector B. Notice that the component perpendicular to vector B has a magnitude of |A|sin θ THEREFORE when you find the CROSS PRODUCT, the result is: i) The MAGNITUDE of one vector, in this case |B| and, ii) The MAGNITUDE of the 2nd vector's component that runs perpendicular to the first vector. ( that is where the sine comes from)

Cross Products in Physics There are many cross products in physics. You will see the matrix system when you learn to analyze circuits with multiple batteries. The cross product system will also be used in mechanics (rotation) as well as understanding the behavior of particles in magnetic fields. A force F is applied to a wrench a displacement r from a specific point of rotation (ie. a bolt). Common sense will tell us the larger r is the easier it will be to turn the bolt. But which part of F actually causes the wrench to turn? |F| Sin θ

Cross Products in Physics What about the DIRECTION? Which way will the wrench turn? Counter Clockwise Is the turning direction positive or negative? Positive Which way will the BOLT move? IN or OUT of the page? OUT You have to remember that cross products give you a direction on the OTHER axis from the 2 you are crossing. So if “r” is on the x-axis and “F” is on the y-axis, the cross products direction is on the z-axis. In this case, a POSITIVE k-hat.

Finding a Cross Product using matrices 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 left to right are subtracted. - - - + + Use your calculator!!!!!! Final answer in vector form.