Question

Determine whether each statement is true or false in double-struck R3. (a) Two lines parallel to a third line are parallel. True False (b) Two lines perpendicular to a third line are parallel. True False (c) Two planes parallel to a third plane are parallel. True False (d) Two planes perpendicular to a third plane are parallel. True False (e) Two lines parallel to a plane are parallel. True False (f) Two lines perpendicular to a plane are parallel. True False (g) Two planes parallel to a line are parallel. True False (h) Two planes perpendicular to a line are parallel. True False (i) Two planes either intersect or are parallel. True False (j) Two lines either intersect or are parallel. True False (k) A plane and a line either intersect or are parallel. True False

Answer 1

Answer:

(a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False

(h) True , (i) True , (j) False , (k) True

Step-by-step explanation:

* Lets explain how to solve the problem

(a) Two lines parallel to a third line are parallel (True)

- Their direction vectors are scalar multiplies of the direction of the 3rd

 line, then they are scalar multiples of each other so they are parallel

(b) Two lines perpendicular to a third line are parallel (False)

- The x-axis and the y-axis are ⊥ to the z-axis but not parallel to

  each other

(c) Two planes parallel to a third plane are parallel (True)

- Their normal vectors parallel to the  normal vector of the 3rd plane,

  so these two normal vectors are parallel to each other and the

  planes are parallel

(d) Two planes perpendicular to a third plane are parallel (False)

- The xy plane and yz plane are not parallel to each other but both

 ⊥ to xz plane

(e) Two lines parallel to a plane are parallel (False)

- The x-axis and y-axis are not parallel to each other but both parallel

  to the plane z = 1

(f) Two lines perpendicular to a plane are parallel (True)

- The direction vectors of the lines parallel to the normal vector of

  the plane, then they parallel to each other , so the lines are parallel

(g) Two planes parallel to a line are parallel (False)

- The planes y = 1 and z = 1 are not parallel but both are parallel to

  the x-axis

(h) Two planes perpendicular to a line are parallel (True)

- The normal vectors of the 2 planes are parallel to the direction of

  line, then they are parallel to each other so the planes are parallel

(i) Two planes either intersect or are parallel (True)

(j) Two lines either intersect or are parallel (False)

- They can be skew

(k) A plane and a line either intersect or are parallel (True)

- They are parallel if the normal vector of the plane and the direction

  of the line are ⊥ to each other , otherwise the line intersect the plane

  at the angle 90° - Ф

Answer 2

Answer:

a) True

(b) False

(c) True

(d) False

(e) False

(f) True  

(g) False

(h) True

(i) True

(j) False

(k) True

Step-by-step explanation:

(a) Two lines parallel to a third line are parallel  

• Their direction vectors are scalar multiplies of the direction of the 3rd  line, then they are scalar multiples of each other so they are parallel

(b) Two lines perpendicular to a third line are parallel  

• The x-axis and the y-axis are ⊥ to the z-axis but not parallel to  each other

(c) Two planes parallel to a third plane are parallel  

• Their normal vectors parallel to the  normal vector of the 3rd plane, so these two normal vectors are parallel to each other and the  planes are parallel.

(d) Two planes perpendicular to a third plane are parallel  

• The xy plane and yz plane are not parallel to each other but both perpendicular  to xz plane

(e) Two lines parallel to a plane are parallel  

• The x-axis and y-axis are not parallel to each other but both parallel to the plane z = 1

(f) Two lines perpendicular to a plane are parallel  

• The direction vectors of the lines parallel to the normal vector of  the plane, then they parallel to each other , so the lines are parallel .

(g) Two planes parallel to a line are parallel  

• The planes y = 1 and z = 1 are not parallel but both are parallel to  the x-axis

(h) Two planes perpendicular to a line are parallel  

• The normal vectors of the 2 planes are parallel to the direction of  line, then they are parallel to each other so the planes are parallel

(i) Two planes either intersect or are parallel  

(j) Two lines either intersect or are parallel  

They can be skew

(k) A plane and a line either intersect or are parallel . They are parallel if the normal vector of the plane and the direction  of the line are ⊥ to each other , otherwise the line intersect the plane at the angle $90 - \theta$