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How to Find 3D Coordinates of a Point in a 3D Cartesian 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 we count the number of units along the x-axis first, then the y-axis, then the z-axis. Positive directions are the arrow directions of each axis, negative are the opposite.
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: from the x-position, move 4 units in the positive y-direction, so y=4.
4
Step 4: Count the units along the z-axis: from the xy-position, move 4 units in the negative z-direction (opposite the arrow), 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 Coordinate System
A 3D coordinate system uses three perpendicular axes (x, y, z) to locate points in three-dimensional space. Points are defined by ordered triples (x, y, z), where each value represents the distance from the origin (0,0,0) along the respective axis. Positive directions are indicated by axis arrows, negative values mean movement opposite to the arrow.
2
Ordered Triple for 3D Points
The standard order for 3D coordinates is (x, y, z). This means you first measure displacement along the x-axis, then parallel to the y-axis, then parallel to the z-axis to reach the target point from the origin.
3
Positive and Negative Axis Directions
In a right-handed 3D coordinate system, the positive x-axis typically points right, positive y-axis points forward/up-right, and positive z-axis points up. Movement in the opposite direction of an axis arrow is represented by a negative coordinate value.
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