Advertisements
Advertisements
Question
Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.
Advertisements
Solution
Given lines are `(1-x)/(3) = (7y -14)/(λ) = (z -3)/(2) and (7 -7x)/(3λ) = (y - 5)/(1) = (6 -z)/(5)`
Converting them into standard form, we have `(x - 1)/(-3) = ("y" - 2)/((λ/7)) = (z - 3)/(2) and (x-1)/((-3λ/7)) = (y - 5)/(1) = (z -6)/(-5)`
Corresponding d.r.'s are `(-3, λ/7, 2) and ((-3λ)/7, 1, -5)`
Since the angle between the lines is right angle so, cos 90° = `|((-3) ((-3λ)/7) + (λ/7) (1) + (2) (-5))/(sqrt((-3)^2 + (λ/7)^2 + 2^2) sqrt(((-3λ)/7)^2+ 1^2 + (-5)^2)))|`
⇒ 0 = `|(9λ/7 + λ/7 - 10)/(sqrt(λ^2/49 + 13) sqrt((9λ^2)/49 + 26)) |`
Squaring and cross-multiplying
⇒ `(10λ/7 - 10)^2 = 0`
⇒ `(10λ)/(7) = 10`
⇒ λ = 7.
Substituting the value λ, of the lines are ` (x - 1)/(-3) = (y - 2)/(1) = (z - 3)/(2)` = a (let) and `(x-1)/(-3) = (y - 5)/(1) = (z -6)/(-5)` = b (let)
From first equation, `(x, y, z) = ( -3a + 1 ,a + 2, 2a+ 3) and "from second equation", (x, y, z) = (-3b + 1, b + 5, -5b + 6)`
Equating the corresponding values of coordinates, we have
- 3a + 1 = - 3b + 1, a + 2 = b + 5 and 2a + 3 = -5b + 6
Or, - 3a + 3b = 0, a - b = 3 and 2a + 5b = 3
Solving the second and third equations of the above, we get a `= (18)/(7)` and b` = (-3)/(7)`
Substituting these values of a and b in the first one
`-3 (18/7) + 3 (-3)/(7) = -9`
Thus, it is clear that the first equation is not satisfied so the lines are not intersecting.
RELATED QUESTIONS
Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines.
`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`
If the Cartesian equations of a line are ` (3-x)/5=(y+4)/7=(2z-6)/4` , write the vector equation for the line.
Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by `(x+3)/3 = (y-4)/5 = (z+8)/6`.
Find the vector equation of the line passing through the points (−1, 0, 2) and (3, 4, 6).
Find the vector equation of a line which is parallel to the vector \[2 \hat{i} - \hat{j} + 3 \hat{k}\] and which passes through the point (5, −2, 4). Also, reduce it to cartesian form.
Find the direction cosines of the line \[\frac{4 - x}{2} = \frac{y}{6} = \frac{1 - z}{3} .\] Also, reduce it to vector form.
Show that the lines \[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} \text { and }\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\] are perpendicular to each other.
Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).
Find the equation of the line passing through the point \[\hat{i} + \hat{j} - 3 \hat{k} \] and perpendicular to the lines \[\overrightarrow{r} = \hat{i} + \lambda\left( 2 \hat{i} + \hat{j} - 3 \hat{k} \right) \text { and } \overrightarrow{r} = \left( 2 \hat{i} + \hat{j} - \hat{ k} \right) + \mu\left( \hat{i} + \hat{j} + \hat{k} \right) .\]
Show that the lines \[\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} \text{ and } \frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5}\] intersect. Find their point of intersection.
Prove that the line \[\vec{r} = \left( \hat{i }+ \hat{j }- \hat{k} \right) + \lambda\left( 3 \hat{i} - \hat{j} \right) \text{ and } \vec{r} = \left( 4 \hat{i} - \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{k} \right)\] intersect and find their point of intersection.
Determine whether the following pair of lines intersect or not:
\[\overrightarrow{r} = \left( \hat{i} - \hat{j} \right) + \lambda\left( 2 \hat{i} + \hat{k} \right) \text{ and } \overrightarrow{r} = \left( 2 \hat{i} - \hat{j} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\]
Find the foot of the perpendicular drawn from the point \[\hat{i} + 6 \hat{j} + 3 \hat{k} \] to the line \[\overrightarrow{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) .\] Also, find the length of the perpendicular
Find the shortest distance between the following pairs of lines whose cartesian equations are: \[\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 5}{5}\]
Find the distance between the lines l1 and l2 given by \[\overrightarrow{r} = \hat{i} + 2 \hat{j} - 4 \hat{k} + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right) \text{ and } , \overrightarrow{r} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k} + \mu\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)\]
Write the cartesian and vector equations of Z-axis.
Write the direction cosines of the line whose cartesian equations are 6x − 2 = 3y + 1 = 2z − 4.
The equations of a line are given by \[\frac{4 - x}{3} = \frac{y + 3}{3} = \frac{z + 2}{6} .\] Write the direction cosines of a line parallel to this line.
The perpendicular distance of the point P (1, 2, 3) from the line \[\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}\] is
If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =
If slopes of lines represented by kx2 - 4xy + y2 = 0 differ by 2, then k = ______
Find the position vector of a point A in space such that `vec"OA"` is inclined at 60º to OX and at 45° to OY and `|vec"OA"|` = 10 units.
Find the cartesian equation of the line which passes ·through the point (– 2, 4, – 5) and parallel to the line given by.
`(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
Find the vector equation of the lines passing through the point having position vector `(-hati - hatj + 2hatk)` and parallel to the line `vecr = (hati + 2hatj + 3hatk) + λ(3hati + 2hatj + hatk)`.
Equation of a line passing through (1, 1, 1) and parallel to z-axis is ______.
For the point \[(5,2,-4)\], what is its position vector \[\vec{a}\]?
A line passes through \[(5,2,-4)\] and is parallel to \[3\hat{i}+2\hat{j}-8\hat{k}\]. Which is its vector equation?
For a line parallel to \[3\hat{i}+2\hat{j}-8\hat{k}\], what are the direction ratios \[a,b,c\]?
Which Cartesian equation is obtained for the line through \[(5,2,-4)\] parallel to \[3\hat{i}+2\hat{j}-8\hat{k}\]?
