Respuesta :
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
This question is based on the properties of lines and planes. Therefore, (a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False
, (h) True , (i) True , (j) False , (k) True.
We have to choose correct statement and marked true or false.
Lets solve the problem.
(a) Two lines parallel to a third line are parallel. (True)
Reason - The direction vectors are scalar multiple of the direction of the third line, then they are scalar multiple of each other. So, they are parallel.
(b) Two lines perpendicular to a third line are parallel. (False)
Reason- As we know that, x-axis and the y-axis are perpendicular to the z-axis but not parallel to each other.
(c) Two planes parallel to a third plane are parallel (True)
Reason- The normal vectors of planes are parallel to the normal vector of the third 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)
Reason- x-y plane and y-z plane are not parallel to each other. But they are perpendicular to x-z plane.
(e) Two lines parallel to a plane are parallel. (False)
Reason - Both x-axis and y-axis are not parallel to each other. But, parallel to the plane z = 1.
(f) Two lines perpendicular to a plane are parallel. (True)
Reason - 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)
Reason- The planes y = 1 and z = 1 are not parallel, but they are parallel to the x-axis.
(h) Two planes perpendicular to a line are parallel. (True)
Reason- The normal vectors of the two planes are parallel to the direction of line. So, they are parallel to each other. Hence, they are parallel.
(i) Two planes either intersect or are parallel (True)
(j) Two lines either intersect or are parallel (False)
Reason- They can also be skew.
(k) A plane and a line either intersect or are parallel (True)
Reason- They are parallel, if the normal vector of the plane and the direction of the line are perpendicular to each other, otherwise the line intersect the plane at the angle 90° [tex]\theta[/tex].
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https://brainly.com/question/24569174
