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What type of geometric shape does the inequality $z > 0$ represent?
What type of geometric shape does the inequality $z > 0$ represent?
The equation that represents the point (0, 0, 0) has only one solution.
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.
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.
The equation ____ does not have a graph in real numbers.
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Match the following equations with their geometric representations:
Match the following equations with their geometric representations:
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What is the length of side 'a' (BC) of the triangle?
What is the length of side 'a' (BC) of the triangle?
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The cosine law applies only when all sides of a triangle are known.
The cosine law applies only when all sides of a triangle are known.
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What angle does cos equal to when it is 0 according to the cosine law?
What angle does cos equal to when it is 0 according to the cosine law?
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The equation x + y + z = 1 represents all points whose coordinates sum up to _____?
The equation x + y + z = 1 represents all points whose coordinates sum up to _____?
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Match the following equations with their geometric meaning:
Match the following equations with their geometric meaning:
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Which side 'b' (AC) of the triangle has a calculated length?
Which side 'b' (AC) of the triangle has a calculated length?
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A 3-dimensional coordinate system divides space into six octants.
A 3-dimensional coordinate system divides space into six octants.
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What do the axes of a 3-dimensional coordinate system represent?
What do the axes of a 3-dimensional coordinate system represent?
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What is the term used to describe the coordinate system that uses three perpendicular axes in three-dimensional space?
What is the term used to describe the coordinate system that uses three perpendicular axes in three-dimensional space?
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In a right-handed coordinate system, the thumb points in the negative z direction.
In a right-handed coordinate system, the thumb points in the negative z direction.
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How is a point in three-dimensional space represented?
How is a point in three-dimensional space represented?
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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 _____.
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 _____.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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What does the term 'right-handed system' imply regarding the coordinate axes?
What does the term 'right-handed system' imply regarding the coordinate axes?
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In three-dimensional coordinates, two points can have the same x and y values but different z values.
In three-dimensional coordinates, two points can have the same x and y values but different z values.
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What is the distance formula for points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?
What is the distance formula for points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?
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Where does the plane defined by the equation x + y + z = 1 intersect the x, y, and z-axes?
Where does the plane defined by the equation x + y + z = 1 intersect the x, y, and z-axes?
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The equation x + y = 4 represents a cylindrical surface with a base radius of 4.
The equation x + y = 4 represents a cylindrical surface with a base radius of 4.
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What type of surface does the equation x + y + z = 0 represent?
What type of surface does the equation x + y + z = 0 represent?
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The equation z = ______ represents all points where the coordinates are (x, y, z), defining a parabolic cylinder.
The equation z = ______ represents all points where the coordinates are (x, y, z), defining a parabolic cylinder.
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Match the following equations with their respective geometric representations:
Match the following equations with their respective geometric representations:
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What geometric shape is described by the equation x^2 + y^2 = 4?
What geometric shape is described by the equation x^2 + y^2 = 4?
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What is the center of a sphere defined by the equation x^2 + y^2 + z^2 = 25?
What is the center of a sphere defined by the equation x^2 + y^2 + z^2 = 25?
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If the variable z is omitted in an equation, the result is a surface parallel to the z-axis.
If the variable z is omitted in an equation, the result is a surface parallel to the z-axis.
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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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Description
Explore the fundamental concepts of coordinate systems and analytical geometry in three-dimensional space. Learn about the representation of points, the distance formula, and the equations for planes and surfaces. This quiz will test your understanding of 3D geometry principles.
