Question

Find the coordinates of the point. The point is located three units behind the \(y z\) -plane, four units to the right of the \(x z\) -plane, and five units above the \(x y\) -plane.

Short answer

The coordinates of the point are \(-3, 4, 5\).

Step-by-step solution

Understanding the planes

In a three-dimensional space, the \(y z\)-plane refers to the plane where all points have an \(x\)-coordinate of 0. Being 'behind' this plane means we move in the negative \(x\)-axis. Therefore, three units behind the \(y z\)-plane corresponds to \(x = -3\). Similarly, the \(x z\)-plane refers to the plane where all points have a \(y\)-coordinate of 0, and 'right' refers to positive \(y\)-direction. Hence, four units to the right of the \(x z\)-plane corresponds to \(y = 4\). Lastly, the \(x y\)-plane is the plane where all points have a \(z\)-coordinate of 0. Being 'above' this plane means moving in the positive \(z\)-direction. Thus, five units above the \(x y\)-plane relates to \(z = 5\).

Formulate the solution

The coordinates of the point are hence given as \((-3, 4, 5)\). These are found by taking the units specified for each dimension/in relation to each respective plane in the problem.

Key concepts

Understanding the yz-plane

In three-dimensional space, the yz-plane is a fundamental concept. It is defined as the plane where the x-coordinate is always zero. This means that any point on the yz-plane has coordinates such that the x-value is 0. It essentially represents the vertical depth in the 3D space without any horizontal movement across the x-axis.
When we say a point is behind the yz-plane, we mean that it is located in the negative direction along the x-axis. For example, if a point is three units behind the yz-plane, it will have an x-coordinate of -3. This indicates its position within the 3D space relative to the yz-plane.
In summary, movements related to the yz-plane affect the x-coordinate, determining whether the point falls to the right (positive) or left (negative) of this plane.

Exploring the xz-plane

The xz-plane is another key plane in 3D geometry. Within this plane, the y-coordinate is constantly zero. Therefore, any point on the xz-plane remains steady at y = 0, signifying no vertical movement along the y-axis.
When a location is specified as being to the right of the xz-plane, it involves position changes in the positive y-axis direction. For instance, a point four units to the right of this plane indicates a y-coordinate of 4. This helps describe the point's lateral location in relation to the xz-plane.
Understanding movements involving the xz-plane is crucial for interpreting how points are positioned in space. It elaborates on how far a point is from this plane along the y-axis.

Defining the xy-plane

The xy-plane represents a horizontal plane where the z-coordinate equals zero in the three-dimensional space. This defines any point on this plane as having z = 0, making it lie flat without any elevation or depression in the z direction.
When we talk about a point being above the xy-plane, it means the point is elevated along the positive z-axis. For a point five units above the xy-plane, it will have a z-coordinate of 5. This lets us understand its vertical positioning regarding the xy-plane.
Conceptualizing the xy-plane and movements above or below it is essential to grasping how points are situated vertically in space. It describes how high or low a point sits in relation to this horizontal reference.