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Find the Coordinates of a Point in a 3D Cartesian Coordinate System
Mathematics
Grade 10 (High School)
Question Content
Write the coordinates of the point shown in the 3D coordinate system.
Correct Answer
b. (3, 4, -4)
Detailed Solution Steps
1
Step 1: Recall the 3D coordinate system rules: coordinates are written in the order (x, y, z), where x is the horizontal axis going forward from the origin, y is the horizontal axis going right from the origin, and z is the vertical axis going up from the origin. Negative values go in the opposite direction of each axis's positive arrow.
2
Step 2: Count the units along the x-axis: the point moves 3 units in the positive x direction, so x=3.
3
Step 3: Count the units along the y-axis: the point moves 4 units in the positive y direction, so y=4.
4
Step 4: Count the units along the z-axis: the point moves 4 units in the negative z direction (down from the origin), so z=-4.
5
Step 5: Combine the values to get the coordinate (3, 4, -4), which matches option b.
Knowledge Points Involved
1
3D Cartesian Coordinate System
A system that uses three perpendicular axes (x, y, z) to define the position of a point in 3-dimensional space. Positions are given as ordered triples (x, y, z), where each value represents the distance from the origin (0,0,0) along the corresponding axis. Positive/negative values indicate direction relative to the origin's arrow for each axis.
2
Ordered Triples for 3D Points
The standard format for 3D coordinates is (x, y, z), where the order of the values is fixed. Changing the order changes the position of the point in space, unlike 2D coordinates which only use two values (x, y).
3
Axis Direction and Sign Conventions
In a right-handed 3D coordinate system (the most common type), the positive x-axis points forward, positive y-axis points right, and positive z-axis points up. Movement in the opposite direction of an axis's positive arrow is represented by a negative coordinate value.
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