Solution Found!
Explain why or why not Determine whether the
Chapter 12, Problem 61E(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) The plane passing through the point (1,1,1) with a normal vector \(\mathbf{n}=\langle 1,2,-3\rangle\) is the same as the plane passing through the point (3,0,1) with a normal vector \(\mathbf{n}=\langle-2,-4,6\rangle\).
(b) The equations x+y-z=1 and -x-y+z=1 describe the same plane.
(c) Given a plane Q, there is exactly one plane orthogonal to Q.
(d) Given a line \(\ell\) and a point \(P_{0}\) not on \(\ell\), there is exactly one plane that contains \(\ell\) and passes through \(P_{0}\).
(e) Given a plane R and a point \(P_{0}\), there is exactly one plane that is orthogonal to R and passes through \(P_{0}\).
(f) Any two distinct lines in \(\mathbf{R}^{3}\) determine a unique plane.
(g) If plane Q is orthogonal to plane R and plane R is orthogonal to plane S, then plane Q is orthogonal to plane S.
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) The plane passing through the point (1,1,1) with a normal vector \(\mathbf{n}=\langle 1,2,-3\rangle\) is the same as the plane passing through the point (3,0,1) with a normal vector \(\mathbf{n}=\langle-2,-4,6\rangle\).
(b) The equations x+y-z=1 and -x-y+z=1 describe the same plane.
(c) Given a plane Q, there is exactly one plane orthogonal to Q.
(d) Given a line \(\ell\) and a point \(P_{0}\) not on \(\ell\), there is exactly one plane that contains \(\ell\) and passes through \(P_{0}\).
(e) Given a plane R and a point \(P_{0}\), there is exactly one plane that is orthogonal to R and passes through \(P_{0}\).
(f) Any two distinct lines in \(\mathbf{R}^{3}\) determine a unique plane.
(g) If plane Q is orthogonal to plane R and plane R is orthogonal to plane S, then plane Q is orthogonal to plane S.
ANSWER:Solution 61E
Step 1 of 7:
a. The plane passing through the point (1, 1, 1) with a normal vector n = 〈1, 2, −3〉 is the same as the plane passing through the point (3, 0, 1) with a normal vector 〈−2, −4, 6〉.
This statement is false
The plane passing through the point (1, 1, 1) with a normal vector n = 〈1, 2, −3〉 is given by
And the plane passing through the point (3, 0, 1) with a normal vector 〈−2, −4, 6〉 is
So,both are not same
So,this statement is false