Advertisements
Advertisements
Question
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)`
Options
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Advertisements
Solution
Both A and R are true and R is the correct explanation of A.
Explanation:
The equation of the x-axis may be written as `vecr = thati`. Hence, the acute angle θ between the given line and the x-axis is given by cosθ = `(|1 xx 1 + (-1) xx 0 + 0 xx 0|)/(sqrt(1^2 + (-1)^2 + 0^2) xx sqrt(1^2 + 0^2 + 0^2)) `
= `1/sqrt(2)`
⇒ θ = `π/4`
APPEARS IN
RELATED QUESTIONS
Find the angle between the following pair of lines:
`vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)`
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 lines whose direction ratios are a, b, c and b − c, c − a, a − b.
Find the angle between the line \[\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}\] and the plane 2x + y − z = 4.
Show that the line whose vector equation is \[\vec{r} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i} + 3 \hat{j} + 4 \hat{k} \right)\] is parallel to the plane whose vector \[\vec{r} \cdot \left( \hat{i} + \hat{j} - \hat{k} \right) = 7 .\] Also, find the distance between them.
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.
State when the line \[\vec{r} = \vec{a} + \lambda \vec{b}\] is parallel to the plane \[\vec{r} \cdot \vec{n} = d .\]Show that the line \[\vec{r} = \hat{i} + \hat{j} + \lambda\left( 3 \hat{i} - \hat{j} + 2 \hat{k} \right)\] is parallel to the plane \[\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .\] Also, find the distance between the line and the plane.
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 lines whose direction cosines are given by the equations: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
`vecr = 3hati + hatj + 2hatk + l(hati - hatj + 2hatk)` and `vecr = 2hati + hatj + 56hatk + m(3hati - 5hatj + 4hatk)`
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 ______.
For direction ratios \[(a_1,b_1,c_1)\] and \[(a_2,b_2,c_2)\], which expression gives \[\cos\theta\]?
If \[(l_1,m_1,n_1)\] and \[(l_2,m_2,n_2)\] are direction cosines of two lines, then \[\cos\theta\] equals:
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:
Two lines with direction ratios \[(a_1,b_1,c_1)\] and \[(a_2,b_2,c_2)\] are parallel when:
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:
If two lines do not pass through the origin, what should be imagined to determine the angle between them?
