Vectors for 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 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 3 iˆ 5 ˆj 8kˆ 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 ( ) ( ) A A 2 ( A ) 2 x 2 y z A = 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: ( ) 2 ( ) 2 3 5 ( 8) 2 = 9. 899 What about the direction? In Physics 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.
A Vector Addition If you define vectors A and as: Then: = = A iˆ x iˆ x A y y ˆj ˆj A kˆ r r A = ( A ) iˆ ( A ) ˆj ( A ) kˆ z z kˆ x x y y z z
Example of Vector Addition If you define vectors A and as: A = 3 iˆ 5 ˆj 8kˆ = 2iˆ 4 ˆj 7kˆ r r A = (3 2) iˆ ( 5 4) ˆj (8 ( 7)) kˆ r r A = 5iˆ 1ˆj 1kˆ Note: Answer is vector
How many combinations of components can a vector have?
What s happening here?
Vector Multiplication A A Dot Product Cross Product Also known as a scalar product. Measure of dependency of A and Mag of A and component of parallel to A are multiplied " Also known as a vector product. " Measure of independency of A and A " Mag of A and component of perpendicular to A are multiplied
A Finding a Dot Product If you define vectors A and as: = = A iˆ x iˆ x Then: A y y ˆj ˆj A kˆ z z kˆ Where A x and x are the x-components, A y and y are the y-components, A z and z are the z-components. A = A A x x y y A z z Answer is a Scalar only, no i, j, k unit vectors.
Example of Dot Product If you define vectors A and as: A = 3 iˆ 5 ˆj 8kˆ = 2iˆ 4 ˆj 7kˆ A = ( 5) 4 8 ( 7) 3 2 A = 6 20 56 = 70 Note: Answer is Scalar 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 = A cosθ Where A & are the magnitudes of the corresponding vectors and θ is the angle between them. A θ
Using a Dot Product in Physics Remember in Physics 1 To calculate Work W = F d cosθ Where F is force, d is displacement, and θ is the angle between the two. Now with calculus: W = F d Note: This symbol means anti-derivative we will learn this soon Dot product of 2 vector quantities
Right Hand Rule and Cross Product - What does it mean?
What is the direction of A X?
Finding a Cross Product 3D If you define vectors A and as: A Then: = = A iˆ x iˆ x A y y ˆj ˆj A kˆ z z kˆ Where A x and x are the x-components, A y and y are the y-components, A z and z are the z-components. A = iˆ Ax x ˆj Ay y kˆ Az z Answer will be in vector (i, j, k) format. Evaluate determinant for answer
Find the determinants along with the sign of (-1) row#column#
Example of a Cross Product k j i A ˆ 8 5 ˆ ˆ 3 = k j i ˆ 7 4 ˆ ˆ 2 = 7 4 2 8 5 3 ˆ ˆ ˆ = k j i A If you define vectors A and as: Set up the determinant as follows, then evaluate.
Evalua5ng the Determinant A = iˆ 3 ˆj 5 kˆ 8 iˆ 3 ˆj 5 2 4 7 2 4 A = iˆ ( 5 7) ˆ(8 j 2) kˆ(3 4) kˆ( 5 2) iˆ(8 4) ˆ(3 j 7) Final answer in vector form. A = ( 3)ˆ i (37) ˆj (22) kˆ
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. A = Where A & are the magnitudes of the corresponding vectors and θ is the angle between them. A θ Asinθ 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 righthand rule
Using a Cross Product in Physics Remember in Physics 1 To calculate Torque τ = l F sinθ Where F is force, l is lever-arm, and θ is the angle between the two. τ = l F
When will the Torque be more? Cross product significance
Some interesting facts A = A The commutative property applies to dot products but not to cross products. A = ( ) A Doing a cross product in reverse order will give the same magnitude but the opposite direction
110 deg Problems
In the figure, vector a lies in the xy plane, has a magnitude of 18 units and points in a direction 250 from the positive direction of the x axis. Also, vector b has a magnitude of 12 units and points in the positive direction of the z axis. What is the vector product = a b? 216 @ 160 deg