Advertisements
Advertisements
Question
Find the angle between the two lines `2x = 3y = -z and 6x =-y = -4z`
Solution
2x = 3y = -z
`x/3 = y/2 = Z/-6`
and 6x = -y = -4z
`cos theta = |(a_1a_2 + b_1b_2 + c_1c_2)/sqrt(a_1^2 + b_ )|`
= `|(3(4)+2(-24)+ (-6)-6)/(sqrt((3^2+ 2^2+6^2)) .sqrt((4^2 +24^2 + 6^2 )))|`
`=|(12-48+36)/(sqrt(9+4+36).sqrt(16+576+36)) |`
= 0
`theta =90°`
Angle between the two lines 2x = 3y = -z and 6x = -y =-74z is 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 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 pair of lines:
`vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`
Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b.
The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and –2, 1, 2 is ______.
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.
Find the angle between the line joining the points (3, −4, −2) and (12, 2, 0) and the plane 3x − y + z = 1.
The line \[\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right)\] is parallel to the plane \[\vec{r} \cdot \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 4 .\] Find m.
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.
Find the angle between the line
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.
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 = 2hati - 5hatj + hatk + lambda(3hati + 2hatj + 6hatk)` and `vecr = 2hati - 5hatj + hatk + lambda(3hati + 2hatj + 6hatk)`
`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 the lines 2x = 3y = – z and 6x = – y = – 4z is ______.
