Advertisements
Advertisements
प्रश्न
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.
उत्तर
\[\text{ The given line is parallel to the vector } \vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k} \text{ and the given plane is normal to the vector } \vec{n} = \hat{i} + \hat{j} + 0 \hat{k} . \]
\[\text{ We know that the angle } \theta \text{ between the line and the plane is given by } \]
\[\sin \theta = \frac{\vec{b} . \vec{n}}{\left| \vec{b} \right| \left| \vec{n} \right|}\]
\[ = \frac{\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right) . \left( \hat{i} + \hat{j} + 0 \hat{k} \right)}{\left| \hat{i}+ 2 \hat{j} + 2 \hat{k} \right| \left| \hat{i} + \hat{j} + 0 \hat{k} \right|} = \frac{1 + 2 + 0}{\sqrt{1 + 4 + 4} \sqrt{1 + 1 + 0}} = \frac{3}{3 \sqrt{2}} = \frac{1}{\sqrt{2}}\]
\[ \Rightarrow \theta = \sin^{- 1} \left( \frac{1}{\sqrt{2}} \right) = {45}^o \]
APPEARS IN
संबंधित प्रश्न
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-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`
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 \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j} + 9 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)\] 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.
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.
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.
Find the angle between the line
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: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 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 = 2hati - 5hatj + hatk + lambda(3hati + 2hatj + 6hatk)` and `vecr = 2hati - 5hatj + hatk + lambda(3hati + 2hatj + 6hatk)`
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 ______.
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)`
