Problem 9 - Calculus an Applied Approach

Question

Find the coordinates of the point. $$ \begin{array}{l}{\text { The point is located on the } x \text { -axis, } 10 \text { units in front of the }} \\ {y z \text { -plane. }}\end{array} $$

Step-by-Step Solution

Answer

The coordinates of the point are (10, 0, 0).

Step 1: Identify Y and Z coordinates

The point is located on the x-axis. This implies that the values of the y and z coordinates will be zero.

Step 2: Identify X coordinate

The point is specifically located 10 units in front of the yz-plane. Since the yz-plane intersects with the x-axis at x=0, this means that the point is located at x=10 units along the x-axis in the positive direction.

Step 3: Construct the coordinates

Now that we have established the values for x, y and z, we can write down the coordinates using the format (x, y, z).

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a fundamental framework in mathematics and physics used to identify the position of a point in space. It consists of perpendicular axes, usually labeled as the x, y, and z axes, which intersect at a point called the origin. To specify the location of a point within this three-dimensional system, we provide coordinates in the form of \( (x, y, z) \), where \( x \) denotes the position along the x-axis, \( y \) along the y-axis, and \( z \) along the z-axis. You can visualize this system like a three-dimensional grid that stretches infinitely in all directions.

Understanding the Cartesian system is crucial for students as it is used in a wide range of applications from graphing simple equations to navigating complex spaces in engineering and virtual environments. Points on this grid are determined by moving a certain distance from the origin along these axes, exactly like a treasure map where you take steps north, east, and up or down to reach your treasure.

X-axis Points

Points located on the x-axis are unique in the Cartesian coordinate system because their y and z coordinates are both zero. This means that any point solely on the x-axis has motioned to the left or right from the origin but has not moved up, down, forwards, or backwards. Hence, coordinates for points on the x-axis take the simple form of \( (x, 0, 0) \).

In the context of our exercise, when a point is said to be '10 units in front of the yz-plane,' it implies that the point is moved 10 units positively along the x-axis - to the right if you're imagining standing at the origin looking along the positive direction of the x-axis. This is because the yz-plane is effectively a vertical 'wall' that stands directly on the x-axis at the point where \( x = 0 \) and extends infinitely upwards and downwards.

YZ-Plane Intersection

The yz-plane is one of the three fundamental planes in three-dimensional space used in the Cartesian coordinate system. It's the plane that stretches up and down (along the y-axis) and forwards and backwards (along the z-axis) – but has no thickness to the left or right. This plane intersects the x-axis at a single point: the origin \( (0, 0, 0) \).

When the exercise mentions '10 units in front of the yz-plane,' it indicates a distance measured from the yz-plane's intersection with the x-axis, which is the origin. Therefore, to move 'in front of' the plane from this intersection point is to move along the positive x-axis. This directional terminology is often used in spatial descriptions, where 'in front of' the yz-plane typically implies the positive x-axis region, whereas 'behind' the yz-plane would imply the negative x-axis region.