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Chapter 1: Problem 22

Find the (parametric) equation of the line \(L\) (a) through the points \(P(1,3,2)\) and \(Q(2,5,-6)\) (b) containing the point \(P(1,-2,4)\) and perpendicular to the plane \(H\) given by the equation \(3 x+5 y+7 z=15\)

Short Answer

Expert verified
(a) The parametric equation for the line L through points P(1,3,2) and Q(2,5,-6) is: \( x = 1 + 1t \) \( y = 3 + 2t \) \( z = 2 - 8t \) (b) The parametric equation for the line L containing the point P(1,-2,4) and perpendicular to the plane H is: \( x = 1 + 3t \) \( y = -2 + 5t \) \( z = 4 + 7t \)

Step by step solution

01

1. Find the direction vector of the line L

To find the direction vector in this case, we can simply subtract the coordinates of point P from the coordinates of point Q. Direction vector = Q - P Direction vector = \( (2 - 1, 5 - 3, -6 - 2) \) Direction vector = (1, 2, -8)
02

2. Write the parametric equation of the line L

Using the point P(1, 3, 2) and the direction vector (1, 2, -8), the parametric equation for line L can be written as: \( x = 1 + 1t \) \( y = 3 + 2t \) \( z = 2 - 8t \) Where t is the parameter. Answer for case (a): The parametric equation for the line L through points P(1,3,2) and Q(2,5,-6) is: \( x = 1 + 1t \) \( y = 3 + 2t \) \( z = 2 - 8t \) For case (b):
03

1. Find the direction vector perpendicular to the plane H

The direction vector perpendicular to the plane 3x + 5y + 7z = 15 can be found by taking the coefficients of x, y and z in the plane equation. Direction vector = (3, 5, 7)
04

2. Write the parametric equation of the line L

Using the point P(1, -2, 4) and the direction vector (3, 5, 7), the parametric equation for line L can be written as: \( x = 1 + 3t \) \( y = -2 + 5t \) \( z = 4 + 7t \) Where t is the parameter. Answer for case (b): The parametric equation for the line L containing the point P(1,-2,4) and perpendicular to the plane H is: \( x = 1 + 3t \) \( y = -2 + 5t \) \( z = 4 + 7t \)

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Most popular questions from this chapter

Find the equation of the plane \(H:\) (a) with normal \(\mathbf{N}=3 \mathbf{i}-4 \mathbf{j}+5 \mathbf{k}\) and containing the point \(P(1,2,-3)\) (b) parallel to \(4 x+3 y-2 z=11\) and containing the point \(Q(2,-1,3)\)

Let \(W\) denote the subspace of \(R^{5}\) consisting of all the vectors having coordinates that sum to zero. The vectors $$ \begin{array}{ll} u_{1}=(2,-3,4,-5,2), & u_{2}=(-6,9,-12,15,-6), \\ u_{3}=(3,-2,7,-9,1), & u_{4}=(2,-8,2,-2,6), \\ u_{5}=(-1,1,2,1,-3), & u_{6}=(0,-3,-18,9,12), \\ u_{7}=(1,0,-2,3,-2), & u_{8}=(2,-1,1,-9,7) \end{array} $$ generate W. Find a subset of the set $\left\\{u_{1}, u_{2}, \ldots, u_{8}\right\\}$ that is a basis for W.

Find the vector \(v\) identified with the directed line segment \(P Q\) for the points: (a) \(P(2,3,-7)\) and \(Q(1,-6,-5)\) in \(\mathbf{R}^{3}\) (b) \(P(1,-8,-4,6)\) and \(Q(3,-5,2,-4)\) in \(\mathbf{R}^{4}\)

Find a parametric representation of the line in \(\mathbf{R}^{4}\) that: (a) passes through the points \(P(1,2,1,2)\) and \(Q(3,-5,7,-9)\) (b) passes through \(P(1,1,3,3)\) and is perpendicular to the hyperplane \(2 x_{1}+4 x_{2}+6 x_{3}-8 x_{4}=5\)

Let \(u\) and \(v\) be distinct vectors in a vector space \(\mathrm{V}\). Show that \(\\{u, v\\}\) is linearly dependent if and only if \(u\) or \(v\) is a multiple of the other.
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