A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms.

A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

A line is drawn in the direction of and it passes through a point with position vector Find the equations of the line in the vector as well as Cartesian forms.

The Cartesian equations of a line are Find the vector equation of the line.

The Cartesian equations of a line are 3x + 1 = 6y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

Find the Cartesian equations of the line which passes through the point (1, 3, -2) and is parallel to the line given by Also, find the vector form of the equations so obtained.

Find the equations of the line passing through the point (1, -2, 3) and parallel to the line Also find the vector form of this equation so obtained.

Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line

Find the equations of the line passing through the point (-1, 3, -2) and perpendicular to each of the lines and

Find the Cartesian and vector equations of the line passing through the point (1, 2, -4) and perpendicular to each of the lines and

Prove that the lines and intersect each other and find the point of their intersection.

Show that the lines and intersect each other. Also, find the point of their intersection.

Show that the lines and do not intersect each other.

Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line Also, find the length of the perpendicular from the given point to the line.

Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line

Find the vector and Cartesian equations of the line passing through the points A(3, 4, -6) and B(5, -2, 7).

Find the vector and Cartesian equations of the line passing through the points A(2, -3, 0) and B(-2, 4, 3).

Find the vector and Cartesian equations of the line joining the points whose position vectors are and

Find the vector equation of a line passing through the point A(3, -2, 1) and parallel to the line joining the points B(-2, 4, 2) and C(2, 3, 3). Also, find the Cartesian equations of the line.

Find the vector equation of a line passing through the point having the position vector and parallel to the line joining the points with position vectors and Also, find the Cartesian equivalents of this equation.

Find the coordinates of the foot of the perpendicular drawn from the point A(1, 2, 1) to the line joining the points B(1, 4, 6) and C(5, 4, 4).

Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).

Find the image of the point (0, 2, 3) in the line

Find the image of the point (5, 9, 3) in the line

Find the image of the point (2, -1, 5) in the line

Show that the points A(2, 1, 3), B(5, 0, 5) and C(-4, 3, -1) are collinear.

Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.

Find the value of λ for which the points A(2, 5, 1), B(1, 2, -1) and C(3, λ, 3) are collinear.

Find the values of λ and μ so that the points A(3, 2, -4), B(9, 8, -10) and C(λ, μ -6) are collinear.

Find the values of λ and μ so that the points A(-1, 4, -2), B(λ, μ 1) and C(0, 2, -1) are collinear.

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Find the angle between each of the following pairs of lines:

and

Show that the lines and are perpendicular to each other.

If the lines and are perpendicular to each other then find the value of λ.

Show that the lines x = - y = 2z and x + 2 = 2y – 1 = - z + 1 are perpendicular to each other.

HINT: The given lines are and

Find the angle between two lines whose direction ratios are

i. 2, 1, 2 and 4, 8, 1

ii. 5, -12, 13 and -3, 4, 5

iii. 1, 1, 2 and

iv. a, b, c and (b – c), (c – a), (a – b)

If A(1, 2, 3), B(4, 5, 7), C(-4, 3, -6) and D(2, 9, 2) are four given points then find the angle between the lines AB and CD.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Find the shortest distance between the given lines.

Compute the shortest distance between the lines and Determine whether these lines intersect or not.

Show that the lines and do not intersect.

Show that the lines and intersect.

Also, find their point of intersection.

Show that the lines and intersect.

Also, find their point of intersection.

Find the shortest distance between the lines L₁ and L₂ whose vector equations are

and

HINT: The given lines are parallel.

Find the distance between the parallel lines L₁ and L₂ whose vector equations are and

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line Also, find the distance between these lines.

HINT: The given line is

The required line is

Now, find the distance between the parallel lines L₁ and L₂.

Write the vector equation of each of the following lines and hence determine the distance between them :

and

HINT: The given lines are

Now, find the distance between the parallel lines L₁ and L₂.

Write the vector equation of the following lines and hence find the shortest distance between them :

and

Find the shortest distance between the lines given below:

and

Find the shortest distance between the lines given below:

and

HINT: Change the given equations in vector form.

Find the length and the equations of the line of shortest distance between the lines given by:

and

Find the length and the equations of the line of shortest distance between the lines given by:

and

Find the length and the equations of the line of shortest distance between the lines given by:

and

Find the length and the equations of the line of shortest distance between the lines given by:

and

Show that the lines and intersect and find their point of intersection.

Show that the lines and do not intersect each other.

If a line has direction ratios 2, -1, -2 then what are its direction cosines?

If the equations of a line are find the direction cosines of a line parallel to the given line.

Write the equations of a line parallel to the line and passing through the point (1, -2, 3).

Find the Cartesian equations of the line which passes through the point (-2, 4, -5) and which is parallel to the line

Write the vector equation of a line whose Cartesian equations are

The Cartesian equations of a line are Write the vector equation of the line.

Write the vector equation of a line passing through the point (1, -1, 2) and parallel to the line whose equations are

If P(1, 5, 4) and Q(4, 1, -2) be two given points, find the direction ratios of PQ.

The equations of a line are Find the direction cosines of a line parallel to this line.

The Cartesian equations of a line are Find its vector equation.

Find the vector equation of a line passing through the point (1, 2, 3) and parallel to the vector

The vector equation of a line is Find its Cartesian equation.

Find the Cartesian equation of a line which passes through the point (-2, 4, -5) and which is parallel to the line

Find the Cartesian equation of a line which passes through the point having position vector and is in the direction of the vector

Find the angle between the lines and

Find the angle between the lines and

Show that the lines and are at right angles.

The direction ratios of a line are 2, 6, -9. What are its direction cosines?

A line makes angles 90^o, 135^o and 45^o with the positive directions of x-axis, y-axis and z-axis respectively. what are the direction cosines of the line?

What are the direction cosines of the y-axis?

What are the direction cosines of the vector

What is the angle between the vector and the x-axis?

The direction ratios of two lines are 3, 2, -6 and 1, 2, 2, respectively. The acute angle between these lines is

The direction ratios of two lines are a, b, c and (b – c), (c – a), (a – b) respectively. The angle between these lines is

The angle between the lines and is

If the lines and are perpendicular to each other then k = ?

A line passes through the points A(2, -1, 4) and B(1, 2, -2). The equations of the line AB are

The angle between the lines and is

The angle between the lines and is

A line is perpendicular to two lines having direction ratios 1, -2, -2 and 0, 2, 1. The direction cosines of the line are

A line passes through the point A(5, -2, 4) and it is parallel to the vector The vector equation of the line is

The Cartesian equations of a line are Its vector equation is

A line passes through the point
A(-2, 4, -5) and is parallel to the line The vector equation of the line is

The coordinates of the point where the line through the points A(5, 1, 6) and B(3, 4, 1) crosses the yz-plane is

The vector equation of the x-axis is given by

The Cartesian equations of a lines are What is its vector equation?

The angle between two lines having direction ratios 1, 1, 2 and 4 is

The straight line is

If a line makes angles α, β and γ with the x-axis, y-axis and z-axis respectively then (sin² α + sin² β + sin² γ) = ?

If (a₁, b₁, c₁) and (a₂, b₂, c₂) be the direction ratios of two parallel lines then

If the points A(-1, 3, 2), B(-4, 2, -2) and C(5, 5, λ) are collinear then the value of λ is