Vectors for Physics. AP Physics C

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1 Vectors for Physics AP Physics C

2 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

3 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)

4 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

5 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

6 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

7 Finding the Magnitude So for the example given the magnitude is: ( ) 2 ( ) ( 8) 2 = 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.

8 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

9 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

10 How many combinations of components can a vector have?

11 What s happening here?

12 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

13 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.

14 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 = = 70 Note: Answer is Scalar only

15 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

16 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

17 Right Hand Rule and Cross Product - What does it mean?

18 What is the direction of A X?

19 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

20 Find the determinants along with the sign of (-1) row#column#

21 Example of a Cross Product k j i A ˆ 8 5 ˆ ˆ 3 = k j i ˆ 7 4 ˆ ˆ 2 = ˆ ˆ ˆ = k j i A If you define vectors A and as: Set up the determinant as follows, then evaluate.

22 Evalua5ng the Determinant A = iˆ 3 ˆj 5 kˆ 8 iˆ 3 ˆj 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

23 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

24 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

25 When will the Torque be more? Cross product significance

26 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

27 110 deg Problems

28 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? 160 deg

Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics

Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector