Advertisements
Advertisements
Question
Find the angle between the following two lines:
`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`
`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`
Advertisements
Solution
Angle between two vectors
`vecr = veca_1 + λvecb_1` and `vecr = veca_2 + μvecb_2` is given by
cos θ = `|(vecb_1 . vecb_2)/(|vecb_1||vecb_2|)|`
`vecr = (2hati - 5hatj + hatk) + λ(3hati + 2hatj + 6hatk)`
`veca_1 = 2hati - 5hatj + hatk, vecb_1 = 3hati + 2hatj + 6hatk`
`vecr = (7hati - 6hatk) + μ(hati + 2hatj + 2hatk)`
`veca_2 = (7hati - 6hatk), vecb_2 = hati + 2hatj + 2hatk`
Now `vecb_1 . vecb_2 = (3hati + 2hatj + 6hatk) . (hati + 2hatj + 2hatk)`
= 3 + 4 + 12
= 19
`|vecb_1| = sqrt(3^2 + 2^2 + 6^2)`
= `sqrt(49)`
= 7
`|vecb_2| = sqrt(1^2 + 2^2 + 2^2)`
= `sqrt(9)`
= 3
cos θ = `|b_1 . b_2|/(|b_1||b_2|)`
cos θ = `|19/(7 xx 3)|`
= `19/21`
θ = `cos^-1 19/21`
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 = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`
Find the angle between the following pairs of lines:
`x/y = y/2 = z/1` and `(x-5)/4 = (y-2)/1 = (z - 3)/8`
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 \[\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}\] 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.
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.
Show that the plane whose vector equation is \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 1\] and the line whose vector equation is \[\vec{r} = \left( - \hat{i} + \hat{j} + \hat{k} \right) + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right)\] are parallel. Also, find the distance between them.
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.
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)`
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 ______.
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 \[\vec b_1=(1,2,2)\] 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?
If two lines do not pass through the origin, what should be imagined to determine the angle between them?
