The dot product of this perpendicular vector to the plane and any vector that lies in the plane is zero. Then the cross product of the 2 displacement vectors are perpendicular the the plane. We could get 2 displacement vectors from the 3 position vectors. We may use 2 points (position vectors) to define the parametric equation of a line, where r and r0 are both position vectors and u is velocity vector: r = r 0 + u tģ points (position vectors) could define a plane. Use right hand rule to determine the direction of the product: suppose A × B, point your fingers in the direction of A, curl your fingers to the vector B, then the thumb is pointing to the direction of the product. The way to remember what the vector is, is to use a three-by-three determinant. B = a 1 b 1 + a 2 b 2 + a 3 b 3 Cross ProductĬross product is also called vector product.You are given vectors and you are free to choose any coordinate system you want. R 2 - r 1 = (x 2 - x 1) i + (y 2 - y 1) j + (z 2 - z 1) kĭot product is also called scalar product, because the multiplication of 2 vectors results in a scalar. The difference between 2 position vectors. Place the tail of a vector at the origin, and then locate a point in space. There are 2 kind of vectors widely used: Position vector In Cartesian coordinate system, you have 3 dimensions: x, y, z. and will not change the direction of the vectors, but just their length. Multiplication of vectors by a scalar is distributive.Addition of vectors is commutative, and associative.Besides, scalars are quantities that have only length, they are just numbers. It is not anchored in any particular spot. VectorsĪ vector is a quantity that has a length associated with it, and a direction. Calculus 1 and 2 are Differential Calculus and Integral Calculus respectively (both are single variate). But remember to add C.Vector Calculus is also known as Multivariate Calculus or “Calculus 3”. If we are lucky enough to find the function on the result side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. Which teaches us to always remember "+C". And the increase in volume can give us back the flow rate.The flow still increases the volume by the same amount.The derivative of the volume x 2+C gives us back the flow rate:Īnd hey, we even get a nice explanation of that "C" value. The integral of the flow rate 2x tells us the volume of water: Derivative: If the tank volume increases by x 2, then the flow rate must be 2x.Integration: With a flow rate of 2x, the tank volume increases by x 2. Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap):Īs the flow rate increases, the tank fills up faster and faster: This shows that integrals and derivatives are opposites! We can integrate that flow (add up all the little bits of water) to give us the volume of water in the tank. The input (before integration) is the flow rate from the tap. So we wrap up the idea by just writing + C at the end. So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value. and the derivative of x 2+99 is also 2x,īecause the derivative of a constant is zero.and the derivative of x 2+4 is also 2x,.It is there because of all the functions whose derivative is 2x: The symbol for "Integral" is a stylish "S"Īfter the Integral Symbol we put the function we want to find the integral of (called the Integrand),Īnd then finish with dx to mean the slices go in the x direction (and approach zero in width). Integration can sometimes be that easy! Notation That simple example can be confirmed by calculating the area:Īrea of triangle = 1 2(base)(height) = 1 2(x)(2x) = x 2
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |