Points on a line?

Question

Q1) Decide which of the following points lie on the line, x= (3,4,4) + t(-2,1,3)

(3,4,4)

(-2,1,3)

(-1,7,10)

(7,2,-2)

 

Q2) Which points lie on the line, x= (1,1,3) + t(1,-1,-5) ?

(3,-1,-7)

(2,-1,3)

(-4,6,28)

(2,1,-5)

 

 

 

 

Answer 1

Q1).

The line equation is X = (3, 4, 4) + t(- 2, 1, 3).

i.e, (x, y, z ) = (3, 4, 4) + t(- 2, 1, 3)

Solve the equations in terms of t.

.

Rewrite this yields as :

, these are called symmetric equations for the given line.

To decide which of the points is lie on the line X = (3, 4, 4) + t(- 2, 1, 3), substitute each point in the symmetric equations.

• Substitute the point (x, y, z ) = (3, 4, 4) in equation (1).

.

The above statement is true.

So, the point (3, 4, 4) is lie on the line X = (3, 4, 4) + t(- 2, 1, 3).

• Substitute the point (x, y, z ) = (- 2, 1, 3) in equation (1).

.

The above statement is false.

So, the point (- 2, 1, 3) is does not lie on the line X = (3, 4, 4) + t(- 2, 1, 3).

• Substitute the point (x, y, z ) = (- 1, 7, 10) in equation (1).

.

The above statement is false.

So, the point (- 1, 7, 10) is does not lie on the line X = (3, 4, 4) + t(- 2, 1, 3).

• Substitute the point (x, y, z ) = (- 7, 2, - 2) in equation (1).

.

The above statement is true.

So, the point (7 , 2, - 2) is lie on the line X = (3, 4, 4) + t(- 2, 1, 3).

Therefore, the points (3, 4, 4) and (7, 2, - 2) are lie on the line X = (3, 4, 4) + t(- 2, 1, 3).

Answer 2

Contd..........

Q2).

The line equation is X = (1, 1, 3) + t(1, - 1, - 5).

i.e, (x, y, z) = (1, 1, 3) + t(1, - 1, - 5)

Solve the equations in terms of t.

.

Rewrite this yields as :

, these are called symmetric equations for the given line.

To decide which of the points is lie on the line X = (1, 1, 3) + t(1, - 1, - 5), substitute each point in the symmetric equations.

• Substitute the point (x, y, z) = (3, - 1, - 7) in equation (1).

.

The above statement is true.

So, the point (2, - 1, 3) is lie on the line  X = (1, 1, 3) + t(1, - 1, - 5).

• Substitute the point (x, y, z) = (- 2, 1, 3) in equation (1).

.

The above statement is false.

So, the point (2, - 1, 3) is does not lie on the line  X = (1, 1, 3) + t(1, - 1, - 5).

• Substitute the point (x, y, z) = (- 4, 6, 28) in equation (1).

.

The above statement is true.

So, the point (- 4, 6, 28)  is lie on the line  X = (1, 1, 3) + t(1, - 1, - 5).

• Substitute the point (x, y, z) = (2, 1, - 5) in equation (1).

.

The above statement is false.

So, the point (2 , 1, - 5) is does not lie on the line  X = (1, 1, 3) + t(1, - 1, - 5).

Therefore, the points (- 4, 6, 28) and (2, - 1, 3) are lie on the line  X = (1, 1, 3) + t(1, - 1, - 5).