unit vector notation

Transcription

1 Unit Vector Notation n 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.

2 Unit Vectors n unit vector is a vector that points along the x, y or z axis and is one unit long. q The symbols for the x, y, and z unit vectors are î, ĵ, and ˆk q ny vector can be expressed as a sum of its x, y and z components multiplied by unit vectors. q The dimensions of the quantity are stated along with the unit vectors, e.g. v = 3m /s i ˆ + 2m /s ˆ j 4m /s k ˆ

3 Unit Vector Notation j = vector of magnitude 1 in the y direction =3j i = vector of magnitude 1 in the x direction = 4i The hypotenuse in Physics is called the RESULTNT or VECTOR SUM. = 4 iˆ + 3 ˆj 3j The LEGS of the triangle are called the COMPONENTS Vertical Component 4i Horizontal Component NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HED TO TOE.

4 Unit Vector Notation in 3 D iˆ ˆj kˆ - unit vector = 1in the + x direction - unit vector = 1in the + y direction - unit vector = 1in the + z direction 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. How would you write vectors J and K in unit vector notation? J = 2 iˆ + 4 ˆj K = 2iˆ 5 ˆj Read only

5 Operations with Vectors: Dot Product n The dot product is the product of the magnitude of two vectors times the cos of the angle between them. B = Bcosθ n The cos means that only the component of one vector that lies along the direction of the other vector contributes to the product. n Because the dot product is a scalar, it is sometimes called the scalar product.

6 The Dot Product (Vector Multiplication) Multiplying 2 vectors sometimes gives you a SCLR quantity which we call the SCLR DOT PRODUCT. B The dot product between and B produces a SCLR quantity. The magnitude of the scalar product is defined as: θ B cos θ Where θ is the NET angle between the two vectors. s shown in the figure. θ B - B cos θ θ

7 Dot Products in Physics Consider this situation: force F is applied to a moving object as it transverses over a frictionless surface for a displacement, d. s F is applied to the object it will increase the object's speed! But which part of F really causes the object to increase in speed? It is F Cos θ! Because it is parallel to the displacement d In fact if you apply the dot product, you get ( F Cos θ)d, which happens to be defined as "WORK B = W = B F x = cosθ F x cosθ Work is a type of energy and energy DOES NOT have a direction, that is why WORK is a scalar or in this case a SCLR PRODUCT (K DOT PRODUCT).

8 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: = < θ 1 & B = B < θ 2 The cross product between and B produces a VECTOR quantity. The magnitude of the vector product is defined as: B θ Where θ is the NET angle between the two vectors. s shown in the figure.

9 Operations with Vectors: The Cross Product quick review n The cross product yields a vector at right angles to the two vectors. B = Bsinθ The direction of B is given by the Right Hand Rule: Point your thumb in the direction of and your forefinger in the direction of B. Your palm now points in the direction of B.

10 The Vector Cross Product B θ B B = B = 30kˆ sinθ = 12 5 sin150 What about the direction???? Positive k-hat??? We can use what is called the RIGHT HND THUMB RULE. Fingers are the first vector, Palm is the second vector, B Thumb is the direction of the cross product. Cross your fingers,, towards, B so that they CURL. The direction it moves will be either clockwise (NEGTIVE) or counter clockwise (POSITIVE) In our example, the thumb points OUTWRD which is the Z axis and thus our answer would be 30 k-hat since the curl moves counter clockwise.

11 Cross Products and Unit Vectors x B = ˆ i ˆ j The cross product between B and produces a VECTOR of which a 3x3 matrix is need to evaluate the magnitude and direction. ˆ k x y z You start by making a 3x3 matrix with 3 columns, one for i, j, & k- hat. The components then go under each appropriate column. B x B y B z Since a is the first vector it comes first in the matrix

12 Cross Products and Unit Vectors x B = ˆ i ˆ j ˆ k x y z You then make an X in the columns OTHER THN the unit vectors you are working with. For i, cross j x k For j, cross i x k For k, cross i x j B x B y B z IB uses this, I like it! ˆ i ˆ j ˆ k x B = x y z z = y z i ˆ x ˆ j + x y k ˆ B y B z B x B z B x B y B x B y B z Let s start with the i-hat vector: We cross j x k Now the j-hat vector: We cross i x k Now the k-hat vector: We cross i x j iˆ kˆ = ( B y = ( B x z y ) ( B z ) ( B y ˆ j = (B z x ) (B x z ) y x ) )

13 Example Cover row with finger, but reverse order for row j! Let s start with the i-hat vector: We cross j x k Now the j-hat vector: We cross i x k iˆ This is my old way!! = ( 4)(5) ( 6)( 4) = 44 ĵ = ( 6)(3) ( 2)(5) = 8 Now the k-hat vector: We cross i x j k ˆ = ( 2)( 4) ( 4)(3) = 20 The final answer would be: B = 44 iˆ 8 ˆj + 20kˆ

14 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. 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 B = F r = B F r sinθ sinθ

15 Solving particle problems φ vs θ z O r F θ y a) Draw the b) Slide the F to the origin c) draw the torque as a product of Τ = r F F F x

16 Two forces each of 2.0N act on a particle that is 3.0m from an origin. Calculate the force produced by F 1 (on the x) and F 2 (on the y). F 1 F 2 z Τ = r F φ θ r 90 deg to plane! Τ = r F T = 3m x 2N(sin90 ) = 6Nm 30 θ O 150 φ x y 150 deg to plane! Τ = r F T = 3m x 2N(sin150 ) = 3Nm

17 force in the Fx of 2N acts from a position of (0, -4, and 3). The F y =0 and the F z =0N as well. Solve for the torque and the direction ( Cht 11 #19) F 1 z r x 30 θ O y

18 Cross Products in Physics B F r = = B sinθ F r sinθ 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.

19 Cross products using the TI-89 Let s clear the stored variables in the TI-89 B = = F B r = 0.30m = 5N =? Since r is ONLY on the x-axis, it ONLY has an i-hat value. Enter, zeros for the other unit vectors. Using CTLOG, find crossp from the menu. Cross Bx, then xb Look at the answers carefully! Crossing Bx shows that we have a direction on the z-axis. Since it is negative it rotates CW. This means the BOLT is being tightened or moves IN to the page. Crossing xb, causes the bolt to loosen as it moves OUT of the page

20 Operations with Vectors: Derivatives n vector in component form is just a sum of vectors in the x, y and z directions. So, derivatives just follow the rule for sums: Find the derivative of each term separately. n Ex.: r=t+ 5i 3t 2 j r v d = =5+ i 6 tj dt 2 v r a = d = d = 6j dt 2 dt