Coordinate Systems and Geometry in 3D

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Questions and Answers

What type of geometric shape does the inequality $z > 0$ represent?

  • A plane in 3D space
  • A solid sphere
  • A line on the x-axis
  • All points above the x-y plane (correct)

The equation that represents the point (0, 0, 0) has only one solution.

True (A)

Describe what the inequality that states the distance squared from (x, y, z) to (x, y, 0) is at least 4 represents.

It represents all points on or outside the surface of a cylinder with radius 2.

The equation ____ does not have a graph in real numbers.

<p>x^2 + y^2 + z^2 = -1</p> Signup and view all the answers

Match the following equations with their geometric representations:

<p>x^2 + y^2 + z^2 = 25 = Sphere with radius 5 centered at the origin z &gt; 0 = All points above the x-y plane x^2 + y^2 = 4 = Cylinder with radius 2 x = 0, y = 0, z = 0 = Single point at the origin</p> Signup and view all the answers

What is the length of side 'a' (BC) of the triangle?

<p>59 (A)</p> Signup and view all the answers

The cosine law applies only when all sides of a triangle are known.

<p>True (A)</p> Signup and view all the answers

What angle does cos equal to when it is 0 according to the cosine law?

<p>90˚</p> Signup and view all the answers

The equation x + y + z = 1 represents all points whose coordinates sum up to _____?

<p>1</p> Signup and view all the answers

Match the following equations with their geometric meaning:

<p>z = 0 = Represents the x-y plane z = -2 = Represents a horizontal plane cut at z = -2 x = y = Represents a vertical plane containing the line x = y x + y + z = 1 = Represents a plane where the sum of coordinates equals 1</p> Signup and view all the answers

Which side 'b' (AC) of the triangle has a calculated length?

<p>3 (A)</p> Signup and view all the answers

A 3-dimensional coordinate system divides space into six octants.

<p>False (B)</p> Signup and view all the answers

What do the axes of a 3-dimensional coordinate system represent?

<p>Three perpendicular lines representing the x, y, and z dimensions.</p> Signup and view all the answers

What is the term used to describe the coordinate system that uses three perpendicular axes in three-dimensional space?

<p>Cartesian coordinate system (B)</p> Signup and view all the answers

In a right-handed coordinate system, the thumb points in the negative z direction.

<p>False (B)</p> Signup and view all the answers

How is a point in three-dimensional space represented?

<p>(x, y, z)</p> Signup and view all the answers

The distance from the origin to point P = (x, y, z) is given by the formula $r = \sqrt{x^2 + y^2 + z^2}$, where $r$ represents the _____.

<p>distance</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Origin = The point where all coordinate axes intersect Axis = A reference line in a coordinate system Point = An exact location in space represented by coordinates Distance = The length of the shortest path between two points in space</p> Signup and view all the answers

What does the term 'right-handed system' imply regarding the coordinate axes?

<p>The positive z-axis points upward when fingers curl from x to y (B)</p> Signup and view all the answers

In three-dimensional coordinates, two points can have the same x and y values but different z values.

<p>True (A)</p> Signup and view all the answers

What is the distance formula for points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?

<p>$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$</p> Signup and view all the answers

Where does the plane defined by the equation x + y + z = 1 intersect the x, y, and z-axes?

<p>(1, 0, 0), (0, 1, 0), (0, 0, 1) (A)</p> Signup and view all the answers

The equation x + y = 4 represents a cylindrical surface with a base radius of 4.

<p>False (B)</p> Signup and view all the answers

What type of surface does the equation x + y + z = 0 represent?

<p>A plane</p> Signup and view all the answers

The equation z = ______ represents all points where the coordinates are (x, y, z), defining a parabolic cylinder.

<p>constant</p> Signup and view all the answers

Match the following equations with their respective geometric representations:

<p>x + y + z = 1 = Plane that intersects axes at (1, 0, 0), (0, 1, 0), (0, 0, 1) x + y = 4 = Line in the xy-plane z = k = Plane parallel to the xy-plane x^2 + y^2 + z^2 = 25 = Sphere with radius 5</p> Signup and view all the answers

What geometric shape is described by the equation x^2 + y^2 = 4?

<p>A circle (B)</p> Signup and view all the answers

What is the center of a sphere defined by the equation x^2 + y^2 + z^2 = 25?

<p>(0, 0, 0)</p> Signup and view all the answers

If the variable z is omitted in an equation, the result is a surface parallel to the z-axis.

<p>True (A)</p> Signup and view all the answers

Flashcards

Cartesian coordinate system (3D)

A coordinate system in three dimensions where axes are perpendicular to each other, with the origin as the intersection point. It's often used to represent points in space.

Point in 3D space

A point in 3D space is represented by three values (x, y, z), where each value corresponds to the distance along a respective axis from the origin.

Distance formula (3D)

The distance between two points in 3D space can be calculated using the Pythagorean theorem extended to three dimensions.

Right triangle (3D)

A right triangle is a triangle where one angle is 90 degrees. In 3D geometry, you can use the distance formula to check if the sides of a triangle form a right angle.

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Projection onto x-y plane

The line passing through a point in 3D space and its projection onto the x-y plane is perpendicular to the x-y plane. This means that the z-coordinate of the projected point is the same as the original point.

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Distance from origin (3D)

The distance between a point in 3D space and the origin can be found using the Pythagorean theorem, with the x, y, and z coordinates acting as the sides of the right triangle formed.

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Right-hand rule (3D)

The right-hand rule helps to visualize the orientation of the axes in a Cartesian coordinate system. It defines a positive direction for the z-axis based on the direction of the x-axis and y-axis.

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Coordinate representation (3D)

A point in 3D space can be represented by its coordinates (x, y, z), where each coordinate indicates its position along the x, y, and z axes respectively.

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Law of Cosines

The Law of Cosines is a fundamental trigonometric equation that links the lengths of a triangle's sides to the cosine of one of its angles. It allows us to determine a triangle's angles when all its sides are known, or to find an unknown side if one angle and its adjacent sides are known.

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Angle γ

In the Law of Cosines, it represents the angle of the triangle that is opposite the side with length 'c'. It's a key element in relating side lengths to angles.

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Plane in 3D space

In 3-dimensional space, a plane divides the space into two half-spaces, just like a line divides a plane into two half-planes.

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Equation in 3D Space

An equation with three variables defines a surface in 3D space. It describes the set of all points (x, y, z) that satisfy the equation.

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Octant in 3D Space

The x-y-z coordinate axes, and the planes they define, divide the 3D space into eight octants.

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Plane where z = 0

A plane where z=0 is the horizontal plane that coincides with the x-y plane. This plane is like a flat surface where the z-coordinate is always zero.

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Plane where z = -2

A plane where z = -2 is a horizontal plane parallel to the x-y plane, but shifted down along the z-axis by 2 units.

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Plane where x = y

A plane where x = y is a vertical plane that includes the line defined by x=y in the x-y plane.

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Equation y = 0 and z = 1

A straight line parallel to the x-axis, located one unit above the x-axis in the three-dimensional space.

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Equation x = 0, y = 0, and z = 0

The point (0, 0, 0), which is the origin of the three-dimensional coordinate system.

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Inequality z > 0

A region in the three-dimensional space where the value of z is greater than zero. It represents all points that are above the x-y plane.

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Inequality (z - 0)^2 >= 4

All points in the three-dimensional space that are a distance of at least 2 units away from the x-y plane.

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Inequality (x - 0)^2 + (y - 0)^2 + (z - 0)^2 <= 25

A sphere centered at the origin of the three-dimensional coordinate system with a radius of 5 units.

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Plane x + y + z = 1

A plane that intersects the x, y, and z axes at the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) respectively. It is parallel to the plane x + y + z = 0, but intersects the origin.

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Equation x^2 + y^2 = 4

A vertical cylinder that contains a circle in the x-y plane. The cylinder's radius is 2 and its axis is the same as the z-axis.

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Equation z = y^2

A parabolic cylinder with its tangent along the y-axis in the x-y plane. It represents all points with coordinates (x, y, z) where the z-coordinate is equal to the square of the y-coordinate.

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Equation x^2 + y^2 + z^2 = 25

A sphere with center at the origin and radius 5. It represents all points (x, y, z) with a constant distance of 5 from the origin.

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Missing Variable in 3D Equations

When a variable is missing from an equation involving x, y, and z, the equation represents a surface parallel to the axis of the missing variable. This surface can be a plane or a cylinder.

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Equation y^2 + (z - 1)^2 = 4

A missing variable in an equation implies that the surface is parallel to the axis of that missing variable. In this case, the equation represents a circle centered at the origin with a radius of 2 in the y-z plane, and in 3D space, it forms a cylinder with its central axis parallel to the x-axis at a height of 1 (z = 1).

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Equations Without All Variables

An equation without all variables may not represent a 2D or 3D shape, but could be a line, curve, a single point, or even nothing at all. The specific shape depends on the equation's form and constraints.

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Equation x + y + z = 0

The equation represents a plane. It does not represent any specific curve or shape. Therefore, the graph is simply a plane.

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Study Notes

Coordinate Systems in Three Dimensions

  • A point in three-dimensional space is represented by three coordinates (x, y, z)
  • The coordinate axes are perpendicular to each other
  • The origin (O) is the point where the axes intersect
  • The right-hand rule is used to define the orientation of the axes (thumb, index, middle finger)

Analytical Geometry in Three Dimensions

  • Points in 3D space are defined using three coordinates (x, y, z).
  • The coordinates specify the position of a point relative to the origin and the axes.
  • Cartesian (rectangular) coordinates are used to represent points in space.
  • The x and y axes define a horizontal plane.
  • The z-axis is vertical.

Distance Formula in 3D

  • The distance (r) between two points P₁ (x₁, y₁, z₁) and P₂ (x₂, y₂, z₂) in three-dimensional space is calculated using the formula: r = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²).

Planes and Surfaces in 3D

  • Equations can represent planes or surfaces in 3D space.
  • A plane in 3D can be defined by an equation, such as x + y + z = 1.
  • This equation represents a plane that intersects the x, y, and z axes at (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively.
  • A cylinder is a surface containing a line segment, called the axis of the cylinder. The equation x² + y² = r² represents a cylinder whose axis is the z-axis and whose radius is r.
  • A sphere is a surface containing a set of points that are all at the same distance from a fixed point. The equation x² + y² + z² = r² represents a sphere whose center is at the origin and whose radius is r.

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