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Essential Calculus: Early Transcendentals
Found in: Page 572
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

(a) Find whether the statement (Two lines parallel to a third line are parallel) is true or false in \({R^3}\).

(b) Find whether the statement (Two lines perpendicular to a third line are parallel) is true or false in \({R^3}\).

(c) Find whether the statement (Two planes parallel to a third plane are parallel) is true or false in \({R^3}\).

(d) Find whether the statement (Two planes perpendicular to a third plane are parallel) is true or false in \({R^3}\).

(e) Find whether the statement (Two lines parallel to a plane are parallel) is true or false in \({R^3}\).

(f) Find whether the statement (Two lines perpendicular to a plane are parallel) is true or false in \({R^3}\).

(g) Find whether the statement (Two planes parallel to a line are parallel) is true or false in \({R^3}\).

(h) Find whether the statement (Two planes perpendicular to a line are parallel) is true or false in \({R^3}\).

(i) Find whether the statement (Two planes either intersect or are parallel) is true or false in \({R^3}\).

(j) Find whether the statement (Two line either intersect or are parallel) is true or false in \({R^3}\).

(k) Find whether the statement (A plane and line either intersect or are parallel) is true or false in \({R^3}\).

(a) The statement is True.

(b) The statement is False.

(c) The statement is True.

(d) The statement is False.

(e) The statement is False.

(f) The statement is True.

(g) The statement is False.

(h) The statement is False.

(i) The statement is True.

(j) The statement is True.

(k) The statement is False.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Explanation.

(a)

\({R^3}\) is the three-dimensional coordinate system which contains \(x - y\)-, and \(z\)-coordinates.

If the two lines are parallel to a third line, the direction vectors of first two lines are parallel to the direction vector of a third line and the direction vectors of first two lines are scalar multiples of direction vector of third line.

As the two line vectors are scalar multiples of third direction vector, the first two direction vectors are also scalar multiples of each other. Therefore, the first two lines are also in parallel with each other.

Hence, the statement is true.

Step 2: Explanation.

(b)

Consider a three-dimensional coordinate system, in which the \(x\) - and \(y\)-axis lines are perpendicular to a \(z\)-axis line. Even though, the \(x\) - and \(y\)-axis lines are perpendicular to each other. But not the \(x\) - and \(y\)-axis lines are in parallel with the \(z\)-axis.

Therefore, the two lines perpendicular to a third line are not in parallel with each other.

Hence, the statement is false.

Step 3: Explanation.

(c)

If the two planes are parallel to a third plane, the normal vector of first two planes is parallel to the normal vector of a third plane.

As the two normal vectors are parallel to each other the first two planes are parallel to each other.

Hence, the statement is true.

Step 4: Explanation.

(d)

Consider a three-dimensional coordinate system, in which the \(xy - \) and \(yz - \)planes are perpendicular to the \(xz - \) plane. Even though, the \(xy - \)planes and \(yz - \)planes are perpendicular to each other.

But not the \(xy - \) and \(yz - \)planes are in parallel with the \(Xz - \) plane.

Therefore, the two planes perpendicular to a third plane are not in parallel with each other.

Hence, the statement is false.

Step 5: Explanation.

(e)

Consider a three-dimensional coordinate system, in which the \(x\) - and \(y\)-axis lines are parallel to a plane \(z = 1\). Even though, the \(x\)-and \(y\)-axis lines are perpendicular to each other. But not the \(x\) - and \(y\)-axis lines are in parallel with that plane.

Therefore, the two lines parallel to a plane are not in parallel with each other.

Hence, the statement is false.

Step 6: Explanation.

(f)

If the two lines are perpendicular to a plane, the direction vectors of first two lines are parallel to the normal vector of a plane.

As the direction vectors of first two lines are parallel to the normal vector of a plane, the direction vectors of first two lines also in parallel with each other.

Therefore, the first two lines are in parallel with each other.

Hence, the statement is true.

Step 7: Explanation.

(g)

Consider a three-dimensional coordinate system, in which the planes \(y = 1\) and \(z = 1\) are parallel to the \(x\) axis line. Even though, the planes \(y = 1\) and \(z = 1\) are not in parallel with each other.

Therefore, the two planes parallel to a line are not in parallel with each other.

Hence, the statement is false.

Step 8: Explanation.

(h)

If the two planes are perpendicular to a line, the normal vectors of the planes are parallel to the direction vector of the line and the normal vectors of the planes are parallel to each other.

As the normal vectors of the planes are parallel to each other, the two planes are parallel to each other.

Hence, the statement is true.

Step 9: Explanation.

(i)

The two planes are parallel if their normal vectors are in parallel.

If the two planes are not in parallel, then they intersect at a straight line and make an angle which is termed as acute angle between two planes.

Therefore, the two planes either intersect or are parallel.

Hence, the statement is true.

Step 10: Explanation.

(j)

The two lines not only intersect or parallel but also there is a possibility of skew.

Hence, the statement is false.

Step 11: Explanation.

(k)

Consider a normal vector for the plane and direction vector for the line. If the normal vector of a plane is perpendicular to the direction vector of the line, the plane and line are parallel with each other.

The other possibility is that the normal vector of the plane and the direction vector of the line intersect with certain angle.

Therefore, the plane and line either intersect or are parallel.

Hence, the statement is true.

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