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#deck:Math::Rectangular coordinates
#tags:math precalculus rectangular-coordinates
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On a number line, how many numbers locate a point, and what is the number called?	One. It is called the coordinate of the point.	definition
On a number line, what is the point that corresponds to a number called?	The graph of the number. The number is the coordinate of the point.	vocabulary
Why say coordinate rather than x-axis number?	Coordinate works for any axis, any dimension and any system, including polar, whose numbers lie on no axis. And a number on the x-axis is a point (3, 0), while the x-coordinate of (3, 5) is read off the axis without lying on it.	vocabulary
Is the x-coordinate of (3, 5) the point (3, 0) on the x-axis?	No. (3, 0) is the foot of the perpendicular from (3, 5) to the x-axis, a point with two coordinates. The x-coordinate is the number 3 the axis reads there, and it is what equations compute with.	definition
Does every coordinate lie on an axis?	No. Rectangular coordinates do, one axis each; polar coordinates are a distance and an angle. A coordinate is one slot in the tuple that fixes a position, and the axis label only says which slot.	definition
Is abscissa used for a point on a plain number line?	No. Abscissa is the x-coordinate of a point in the plane. On a line the word is coordinate.	vocabulary
In the plane, how many numbers locate a point, and what are they called together?	Two. Together they are the ordered pair (x, y), the coordinates of the point.	definition
Is (x, y) one coordinate or two?	Two: x is the x-coordinate and y is the y-coordinate. The pair is the point's coordinates, plural. Everyday speech says a coordinate for a whole location; the textbooks do not.	definition
What is the x-axis?	The horizontal number line.	definition
What is the y-axis?	The vertical number line.	definition
In the xy-plane, what is the origin O?	The point where the x-axis and the y-axis intersect. Its coordinates are (0, 0).	definition
What does the number of axes of a coordinate system tell you?	The dimension of the space: one axis per dimension and one coordinate per axis. A line has one, the plane two, space three.	definition
What are the coordinates of the origin on a line, in the plane, and in space?	0 on a line, (0, 0) in the plane, (0, 0, 0) in space: one 0 per axis.	definition
Is the origin a special point of the plane?	No. It is whichever point was chosen as 0 on both axes when they were drawn. Its coordinates are (0, 0) by definition.	definition
Must the x-axis and the y-axis use the same scale?	No. Mathematics usually does; an application may use a different scale on each axis.	definition
Which points on the x-axis are positive, and which negative?	Right of O positive, left of O negative.	definition
Which points on the y-axis are positive, and which negative?	Above O positive, below O negative.	definition
What does the arrowhead at the end of an axis mean?	It marks the positive direction of that axis.	definition
Can an axis have arrowheads on both ends?	Yes. Then they mean the line goes on forever, and the numbers along the axis say which way is positive.	definition
Why does it matter which direction of an axis is positive?	It sets the orientation: with x right and y up, the turn from the x-axis to the y-axis is counterclockwise. Flip one axis and every rotation and slope changes sign on the picture.	reasoning
A computer screen has y positive downward. What changes?	The numbers do not; the picture does. Counterclockwise becomes clockwise, so a textbook rotation formula turns the wrong way.	reasoning
What are the two names of the coordinate system built from two perpendicular number lines?	Rectangular coordinate system, or Cartesian coordinate system, after René Descartes.	definition
What is the xy-plane?	The plane in which the x-axis and the y-axis lie.	definition
What other names does the xy-plane go by?	The Cartesian plane, or the coordinate plane.	vocabulary
Why does Axler say the coordinates are called rectangular?	Each point's coordinates are determined by a rectangle: corners at the origin, the point, and its two feet on the axes, with sides |x| and |y|.	vocabulary
What names does Axler prefer to x-axis, y-axis, x-coordinate and y-coordinate?	Horizontal axis and vertical axis; first coordinate and second coordinate. They name the slot, so they survive an axis called t and a point written (y, x).	vocabulary
What two choices fix the scale on an axis?	Where 0 goes, the origin, and where 1 goes, the unit point. Every other label is then determined.	definition
What is genuinely different from rectangular coordinates?	Polar coordinates: the same point named by its distance from the origin and its angle from the x-axis.	vocabulary
What are the coordinate axes?	The x-axis and the y-axis.	definition
In the ordered pair (x, y) of a point, what does x measure?	The signed distance of the point from the y-axis: how far right or left.	signed-distance
In the ordered pair (x, y) of a point, what does y measure?	The signed distance of the point from the x-axis: how far up or down.	signed-distance
What does signed mean in signed distance?	The sign says which side: x > 0 right of the y-axis, x < 0 left of it.	signed-distance
The x-coordinate measures distance from which axis?	The y-axis. Right or left is measured from the vertical line.	signed-distance
What does Larson call the signed distance of a point from an axis?	The directed distance. Two names, one number.	vocabulary
What is the ordered pair (x, y) of a point P called?	The coordinates of P.	definition
How do you plot the point (−3, 1)?	Go 3 units left of O along the x-axis, then 1 unit straight up, and place a dot.	plotting
What does it mean to plot a point?	To place a dot at the location its coordinates give.	plotting
Must the coordinates of a point be whole numbers?	No. (−1.5, −2.5) is a point in quadrant III. The grid lines are only a reading aid.	plotting
What form do the coordinates of a point on the x-axis have?	(x, 0): the y-coordinate is 0.	axes
What form do the coordinates of a point on the y-axis have?	(0, y): the x-coordinate is 0.	axes
A point has y-coordinate 0. Where is it?	On the x-axis.	axes
A point has x-coordinate 0. Where is it?	On the y-axis.	axes
What is another name for the x-coordinate of a point?	The abscissa.	vocabulary
What is another name for the y-coordinate of a point?	The ordinate.	vocabulary
What is the abscissa of a point?	Its x-coordinate.	vocabulary
What is the ordinate of a point?	Its y-coordinate.	vocabulary
Where do the words abscissa and ordinate come from?	Latin, 1600s: abscissa means cut off, ordinata means applied in order. Older than the letters x and y.	vocabulary
Is the abscissa the x-axis?	No. The abscissa is the number x. The line is the x-axis, or the axis of abscissas.	vocabulary
What is the third coordinate of a point in space called, when it has a name?	The applicate, rare in English; kota in Polish, cote in French. Usually just the z-coordinate.	vocabulary
What are abscissa and ordinate called in Polish, French and German?	Odcięta and rzędna; abscisse and ordonnée; Abszisse and Ordinate.	vocabulary
What is the Polish word for coordinate?	Współrzędna, plural współrzędne: współ- is co- and rzędna is ordinate, the Latin word rebuilt from Polish parts. A coordinate system is układ współrzędnych.	vocabulary
What are quadrant, origin and ordered pair in Polish?	Ćwiartka (pierwsza ćwiartka is quadrant I), początek układu współrzędnych, para uporządkowana.	vocabulary
How is the point P whose coordinates are (x, y) usually written and said?	P = (x, y), and simply the point (x, y).	vocabulary
The notation (x, y) has three meanings. What are they?	A point in the plane; the open interval of all t with x < t < y; a vector, x right and y up.	notation
Can (5, 2) be an open interval?	No. An interval (a, b) needs a < b. (5, 2) must be the point.	notation
Does a vector start at the origin?	Not necessarily. A vector is a displacement and can start anywhere. Started at O, the vector (3, 1) ends at the point (3, 1).	notation
What is the vector from P = (2, 3) to Q = (5, 4)?	(5 − 2, 4 − 3) = (3, 1): the coordinates of Q minus those of P.	notation
What are the quadrants?	The four regions the axes divide the plane into, numbered I to IV counterclockwise from the top right.	quadrants
Signs of x and y in quadrant I?	x > 0 and y > 0 (top right).	quadrants
Signs of x and y in quadrant II?	x < 0 and y > 0 (top left).	quadrants
Signs of x and y in quadrant III?	x < 0 and y < 0 (bottom left).	quadrants
Signs of x and y in quadrant IV?	x > 0 and y < 0 (bottom right).	quadrants
A point has x < 0 and y > 0. Which quadrant?	Quadrant II.	quadrants
A point has x > 0 and y < 0. Which quadrant?	Quadrant IV.	quadrants
A point has x < 0 and y < 0. Which quadrant?	Quadrant III.	quadrants
A point has x > 0 and y > 0. Which quadrant?	Quadrant I.	quadrants
Which quadrant is a point on a coordinate axis in?	None. 0 has no sign, and a quadrant is a pair of signs.	quadrants
Which quadrant is (3, −2) in?	Quadrant IV: x > 0, y < 0.	quadrants
Which quadrant is (−2, −3) in?	Quadrant III: x < 0, y < 0.	quadrants
Which quadrant is (−3, 1) in?	Quadrant II: x < 0, y > 0.	quadrants
Which quadrant is (3, 2) in?	Quadrant I: x > 0, y > 0.	quadrants
In which order are the quadrants numbered around the origin?	Counterclockwise: I top right, II top left, III bottom left, IV bottom right.	quadrants
Why does an angle open from the x-axis toward the y-axis, and not the other way?	By definition: a positive turn carries the first axis onto the second. So a small positive angle lands at a point with positive y, and sin θ is positive just above 0°. Clockwise is only a different convention, as on a clock or a compass.	reasoning
Why are the quadrants numbered counterclockwise?	An angle opens from the positive x-axis toward the positive y-axis, which with x right and y up is counterclockwise. Its first quarter turn, 0° to 90°, sweeps quadrant I, and II, III, IV follow the turn.	reasoning
Are (2, 5) and (5, 2) the same point?	No. (2, 5) is 2 right and 5 up; (5, 2) is 5 right and 2 up.	ordered-pair
Why is a quadrant nothing but a pair of signs?	Membership depends on the signs of x and y, never on their sizes: (1, 1) and (1000, 5) share quadrant I.	reasoning
What happens to the quadrant of (x, y) when x is replaced by −x?	The point reflects across the y-axis: I and II swap, III and IV swap.	reasoning
In which quadrants is the product xy positive?	I and III, where x and y have the same sign. Negative in II and IV.	reasoning