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Chapter 1: Problem 24

Find the points of intersection of the line \(x=3+2 t, y=7+8 t, z=-2+t,\) that \(\mathrm{is}, \mathrm{I}(t)=\) \((3+2 t, 7+8 t,-2+t),\) with the coordinate planes.

Short Answer

Expert verified
The intersection points are \((7, 23, 0)\), \(\left(\frac{5}{4}, 0, -\frac{23}{8}\right)\), and \((0, -5, -\frac{7}{2})\).

Step by step solution

01

Identify the Parametric Equations

The line given in parametric form is \((x, y, z) = (3+2t, 7+8t, -2+t)\). Here, \(x=3+2t\), \(y=7+8t\), and \(z=-2+t\). To find the points of intersection with the coordinate planes, we will solve these equations for when \(x\), \(y\), or \(z\) is zero.
02

Find Intersection with the XY-Plane

The XY-plane is defined by \(z = 0\). Set \(-2 + t = 0\) and solve for \(t\): \[ t = 2 \].Substitute \(t = 2\) back into the expressions for \(x\) and \(y\):\[ x = 3 + 2(2) = 7 \]\[ y = 7 + 8(2) = 23 \]Thus, the intersection point with the XY-plane is \((7, 23, 0)\).
03

Find Intersection with the XZ-Plane

The XZ-plane is defined by \(y = 0\). Set \(7 + 8t = 0\) and solve for \(t\): \[ t = -\frac{7}{8} \].Substitute \(t = -\frac{7}{8}\) back into the expressions for \(x\) and \(z\):\[ x = 3 + 2\left(-\frac{7}{8}\right) = \frac{5}{4} \]\[ z = -2 + \left(-\frac{7}{8}\right) = -\frac{23}{8} \]Thus, the intersection point with the XZ-plane is \(\left(\frac{5}{4}, 0, -\frac{23}{8}\right)\).
04

Find Intersection with the YZ-Plane

The YZ-plane is defined by \(x = 0\). Set \(3 + 2t = 0\) and solve for \(t\): \[ t = -\frac{3}{2} \].Substitute \(t = -\frac{3}{2}\) back into the expressions for \(y\) and \(z\):\[ y = 7 + 8\left(-\frac{3}{2}\right) = -5 \]\[ z = -2 + \left(-\frac{3}{2}\right) = -\frac{7}{2} \]Thus, the intersection point with the YZ-plane is \((0, -5, -\frac{7}{2})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Planes
In three-dimensional space, coordinate planes are the fundamental surfaces that help us understand the position of a point relative to the axes. There are three primary coordinate planes: the XY-plane, the XZ-plane, and the YZ-plane. Each of these divides the 3D space into different regions.
  • The **XY-plane** is where the z-coordinate is always zero. This plane extends along the x and y axes and is crucial for examining how an object behaves when it is projected onto a flat 2D surface in 3D space.
  • The **XZ-plane** has a constant y-coordinate of zero. In this plane, motion is only possible along the x and z directions. It plays a vital role in understanding vertical and horizontal movements relative to the y-axis.
  • The **YZ-plane** features a constant x-coordinate of zero. Objects and movements constrained to this plane are only traceable along the y and z axes, presenting another layer of projection within 3D analyses.
When analyzing intersections with lines defined by parametric equations, these planes offer a simple method to solve for specific conditions like making one variable zero and tracing where the other two delineate intersections.
Intersection Points
Intersection points between a line and coordinate planes help us determine where a line passes through these fundamental surfaces. To find these points, we usually set one of the coordinates (either x, y, or z) to zero and solve the corresponding parametric equations.
  • For the **XY-plane**, set z = 0 because this defines the plane. Solve for the parameter that dictates the values of x and y when z is zero.
  • For the **XZ-plane**, set y = 0 and solve the parametric equations to find x and z values.
  • Similarly, for the **YZ-plane**, set x = 0 to uncover the intersection y and z values.
These intersections often reveal critical points in geometry and physics, helping visualize and quantify where and how lines engage with surfaces or volumes. This straightforward method offers insights into spatial relationships and behaviors in three-dimensional space.
Line in 3D Space
A line in 3D space can be expressed using parametric equations, which define each point on the line as a function of a parameter, typically denoted as \(t\). Such equations are of the form:
  • \(x = a + bt\)
  • \(y = c + dt\)
  • \(z = e + ft\)
Here, \(a, c, e\) are the coordinates where the line intersects the origin point (when \(t = 0\)), while \(b, d, f\) determine the direction and nature of the line鈥檚 progression in space.
  • The **direction vector** \((b, d, f)\) suggests how the line moves in response to changes in \(t\).
  • Understanding this setup allows for the exploration of how lines interact with various objects in space, such as determining where lines meet planes by substituting and solving equations.
This parametric approach is highly practical, especially in fields that involve spatial analysis, such as physics, engineering, and computer graphics. It's an essential tool for depicting not just where an object is, but where it might move to, making it fundamental to many practical applications.

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Most popular questions from this chapter

Let \(\mathbf{v}=(2,3) .\) Suppose \(\mathbf{w} \in \mathbb{R}^{2}\) is perpendicular to \(\mathbf{v}\) and that \(\|\mathbf{w}\|=5 .\) This determines \(\mathbf{w}\) up to sign. Find one such \(\mathbf{w}\)

Let \(P\) be the plane defined by the equation \(x+y+z=1 .\) Which of the following points are contained in \(P ?\) (a) (0,0,0) (b) (1,1,-1) (c) (-3,8,-4) (d) (1,2,-3)

Prove the statements. If \(\mathrm{PQR}\) is a triangle in space and \(b>0\) is a number, then there is a triangle with sides parallel to those of \(\mathrm{PQR}\) and side lengths \(b\) times those of \(\mathrm{PQR}\).

Find the angle between \(7 \mathbf{j}+19 \mathbf{k}\) and \(-2 \mathbf{i}-\mathbf{j}\) (to the nearest degree \()\)

Let \(\mathbf{b}=(3,1,1)\) and \(P\) be the plane through the origin given by \(x+y+2 z=0\) (a) Find an orthogonal basis for \(P .\) That is, find two nonzero orthogonal vectors \(\mathbf{v}_{1}, \mathbf{v}_{2} \in P\) (b) Find the orthogonal projection of b onto \(P .\) That is, find \(\operatorname{Proj}_{\mathbf{V}_{1}} \mathbf{b}+\operatorname{Proj}_{\mathbf{V}_{2}} \mathbf{b}\)
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