Essential Mathematics: Coordinate Geometry

17 flashcards · Mathematics · by lakesare

  1. What's the midpoint of the line segment joining the points (x1, y1) and (x2, y2)?

  2. What's a perpendicular bisector of some line segment?

    Perpendicular bisector of a line segment is a line that divides the line segment into two equal parts, and is perpendicular to a line segment. Earlier in this subsection you saw that if you choose any two points, then all the points equidistant from these two points form a straight line that passes through the midpoint of the line segment that joins the two points, and is perpendicular to this line segment.

  3. How would you find the formula of perpendicular bisector given points (a, b) and (c, d). Mark two points on a piece of paper. Draw a third point such that its distance from each of the original points is the same. Then mark some more points with this property.All the points with this property is the perpendicular bisector.

    We can equate the distances from (a, b) to P and from (c, d) to P.

  4. Derive the equation of the circle.

  5. [Finding the equation of a circle] Find the equation of the circle with centre (2, −1) and radius 3.

  6. [Finding the centre and radius of a circle] Find the centre and radius of the circle with equation:

  7. [Identifying the equations of circles] Find whether equation represents a circle, and if so - find the centre and radius.

  8. What's a bearing? If the vector represents the displacement or velocity of an object such as a ship or an aircraft, then its direction is often given as a compass bearing.

    A vector with a bearing of 150°. Notice that the rotational direction in which bearings are measured is opposite to that in which angles are usually measured in mathematics.

  9. What's the position vector of point P? (P is any point is any point in the place or in the three-dimensional space)

    Position vector of P is the vector OP, where O is the origin.

  10. [Magnitude and direction of a vector from its components] Given vector 4i − 3j, find magnitude of the vector, and angle that it makes with the positive x-direction. Give the angle to the nearest degree.

  11. Show what circles contain points A&B.

    All circle with the center on the perpendicular bisector. If you choose a third point, then as long as the three points do not all lie on the same straight line, there is only one circle that passes through all three points.

  12. [Strategy] How to find the center of the circle given 3 points it passes through? Three points A, B and C determine a circle

    Find equations of 2 perpendicular bisectors Find their intersection. The centre of the circle is the point of intersection of the perpendicular bisectors of AB and BC

  13. [Intersection of 2 circles] What's the important step in finding the intersection of 2 circles?

    Solve the usual system of equations - but eliminate the terms x2 and y2 by subtracting one equation from the other! To find the points of intersection of two circles, you need to solve their equations simultaneously. It's not straightforward to use one equation to find an expression for x or y to substitute into the other equation, because both equations have terms in x2 and y2. We should eliminate the terms in x2 and y2 by subtracting one equation from the other (or a multiple of one equation…

  14. Law of demand VS law of supply?

    As price increases: law of demand - consumers want to be less (demand falls) law of supply - producer is willing to offer more products (supply increases) The law of demand states that the lower the selling price of the goods, the more consumers want to buy.The law of supply states that the lower the selling price of the goods, the less producers want to produce.

  15. What axis is usually pointing up in 2-dimensional coordinate system in 3-dimensional coordinate system

    in 2-dimensional CS - y in 3-dimensional CS - z Although two-dimensional coordinate systems are usually drawn with the positive direction of the y-axis pointing up, three-dimensional systems are usually drawn with the positive direction of the z-axis pointing up.

  16. How do the right-handed VS left-handed coordinate system look? The right-handed coordinate system is usually used in preference to the left-handed one.

    Right-handed coordinate system: Left-handed coordinate system: If you do the same with your left hand, then your middle finger points in the positive z-direction of a left-handed coordinate system. The default coordinate system in OpenGL(TM) is right-handed: the positive x and y axes point right and up, and the negative z axis points forward. Positive rotation is counterclockwise about the axis of rotation.

  17. What's the definition of the angle between 2 vectors?

    Angle ≤ 180° when vectors are placed tail-to-tail. The angle between two vectors a and b is the angle θ in the range 0° ≤ θ ≤ 180° between their directions when the vectors are placed tail-to-tail. This definition applies to both 2-dimensional and 3-dimensional vectors.