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How to find the position of a particle at t=a, given parametric equations x(t) and y(t)?
1. Substitute t=a into x(t) to find x-coordinate. 2. Substitute t=a into y(t) to find y-coordinate. 3. Position is (x(a), y(a)).
How to find the velocity vector v(t) given a position vector r(t)=<x(t), y(t)>?
1. Find the derivative of x(t) with respect to t. 2. Find the derivative of y(t) with respect to t. 3. v(t) = <x'(t), y'(t)>.
How to find the speed of a particle at t=a, given v(t)=<x'(t), y'(t)>?
1. Evaluate x'(a) and y'(a). 2. Speed = sqrt((x'(a))^2 + (y'(a))^2).
How to find the acceleration vector a(t) given v(t)=<x'(t), y'(t)>?
1. Find the derivative of x'(t) with respect to t. 2. Find the derivative of y'(t) with respect to t. 3. a(t) = <x''(t), y''(t)>.
How to determine the conic section from its equation?
1. Check for x^2 and y^2 terms. 2. If both are present and have the same coefficient, it's a circle. 3. If both are present and have different coefficients but the same sign, it's an ellipse. 4. If one is present, it's a parabola. 5. If both are present with opposite signs, it's a hyperbola.
How to add two matrices A and B?
1. Ensure A and B have the same dimensions. 2. Add corresponding elements: (A+B)[i,j] = A[i,j] + B[i,j].
How to find the determinant of a 2x2 matrix?
1. For matrix [[a, b], [c, d]], multiply a*d and b*c. 2. Subtract: determinant = ad - bc.
How to parametrize a circle x^2 + y^2 = r^2?
1. Let x = r*cos(t). 2. Let y = r*sin(t).
How to find the sum of two vectors u = <a, b> and v = <c, d>?
1. Add the x-components: a + c. 2. Add the y-components: b + d. 3. Result: u + v = <a+c, b+d>.
How to convert an implicit equation to standard form for a circle?
1. Complete the square for both x and y terms. 2. Rewrite the equation in the form (x-h)^2 + (y-k)^2 = r^2.
Define Parametric Functions.
Functions where x and y coordinates are defined in terms of a parameter, often 't'.
Define Implicitly Defined Functions.
Functions where the relationship between x and y is defined by an equation, not explicitly solved for y.
Define Vectors.
Mathematical objects with both magnitude and direction, often represented as arrows.
Define Matrices.
Rectangular arrays of numbers used to represent linear transformations and solve systems of equations.
Define Conic Sections.
Shapes formed by slicing a cone: circles, ellipses, parabolas, and hyperbolas.
Define Vector-Valued Functions.
Functions that output vectors instead of scalar values, often used to describe the position of a moving object over time.
What is the magnitude of a vector?
The length of the vector.
What is the determinant of a matrix?
A scalar value that indicates if a matrix is invertible.
What is the dot product of two vectors?
A scalar value obtained by multiplying corresponding components of two vectors and summing the results.
What is the cross product of two vectors?
A vector that is perpendicular to both input vectors (defined in 3D space).
What are the differences between Parametric and Implicit Functions?
Parametric: x and y defined by a parameter. | Implicit: x and y related by an equation.
What are the differences between Vectors and Scalars?
Vectors: Have magnitude and direction. | Scalars: Have only magnitude.
What are the differences between Dot Product and Cross Product?
Dot Product: Results in a scalar. | Cross Product: Results in a vector (in 3D).
What are the differences between a circle and an ellipse?
Circle: Constant radius in all directions. | Ellipse: Varying radius (major and minor axes).
What are the differences between a hyperbola and an ellipse?
Ellipse: Sum of distances to foci is constant. | Hyperbola: Difference of distances to foci is constant.
What are the differences between velocity and speed?
Velocity: Vector with magnitude and direction. | Speed: Scalar, magnitude of velocity.
What are the differences between position vector and velocity vector?
Position Vector: Indicates location. | Velocity Vector: Indicates rate of change of position.
What are the differences between matrix addition and matrix multiplication?
Matrix Addition: Element-wise addition, commutative. | Matrix Multiplication: Rows by columns, not commutative.
What are the differences between parametric equations for a line and a circle?
Line: x and y change linearly with t. | Circle: x and y change sinusoidally with t.
What are the differences between explicit and implicit functions?
Explicit: y is isolated on one side (y = f(x)). | Implicit: x and y are intertwined in an equation.