Advertisements
Advertisements
Question
Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b.
Advertisements
Solution
the angle between the given lines is given by
cos θ = `(a(b - c) + b(c - a) + c(a - b))/(sqrt(a^2 + b^2 + c^2). sqrt((a - b)^2 + (b - c)^2 + (c - a)^2)`
cos θ = 0
⇒ Hence, θ = 90°
APPEARS IN
RELATED QUESTIONS
If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 =0,
then show that 100(h2 - ab) = (a + b)2
Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.
Find the angle between the following pairs of lines:
`(x-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`
Find the values of p so the line `(1-x)/3 = (7y-14)/2p = (z-3)/2` and `(7-7x)/(3p) = (y -5)/1 = (6-z)/5` are at right angles.
Find the angle between the line \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j} + 9 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)\] and the plane \[\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 5 .\]
Find the angle between the line joining the points (3, −4, −2) and (12, 2, 0) and the plane 3x − y + z = 1.
Find the angle between the line \[\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}\] and the plane
3x + 4y + z + 5 = 0.
Write the angle between the line \[\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}\] and the plane x + y + 4 = 0.
Find the angle between the two lines `2x = 3y = -z and 6x =-y = -4z`
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.
If l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
`vecr = 3hati + hatj + 2hatk + l(hati - hatj + 2hatk)` and `vecr = 2hati + hatj + 56hatk + m(3hati - 5hatj + 4hatk)`
Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk + λ(hati - hatj)` and the x-axis is `π/4`
Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk + λ(a_1hati + b_1hatj + c_1hatk)` and `barr = x_2hati + y_2hatj + z_2hatk + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`
The angle between two lines `(x + 1)/2 = (y + 3)/2 = (z - 4)/(-1)` and `(x - 4)/1 = (y + 4)/2 = (z + 1)/2` is ______.
A straight line L through the point (3, –2) is inclined at an angle of 60° to the line `sqrt(3)x + y` = 1. If L also intersects the x-axis, then the equation of L is ______.
The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is ______.
Find the angle between the following two lines:
`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`
`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`
For direction ratios \[(a_1,b_1,c_1)\] and \[(a_2,b_2,c_2)\], which expression gives \[\sin\theta\]?
Two lines with direction ratios \[(a_1,b_1,c_1)\] and \[(a_2,b_2,c_2)\] are perpendicular when:
For the lines \[\vec r=\vec a_1+\lambda\vec b_1\] and \[\vec r=\vec a_2+\mu\vec b_2\], the angle between the lines is the angle between:
In the symmetric or Cartesian form \[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\], the direction ratios are:
For \[\vec b_1=(1,2,2)\] and \[\vec b_2=(3,2,6)\], what is \[\vec b_1\cdot\vec b_2\]?
For \[\vec b_1=(1,2,2)\] and \[\vec b_2=(3,2,6)\], which pair gives their magnitudes?
For direction vectors \[\vec b_1=(1,2,2)\] and \[\vec b_2=(3,2,6)\], the angle between the lines is:
What is generally the required angle between two lines?
