Advertisements
Advertisements
प्रश्न
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Advertisements
उत्तर
\[\text { Let } \vec{a} \text{ be a vector parallel to the vector with direction ratios } 2, 3, 6 . \]
\[ \Rightarrow \vec{a} = 2 \hat{i} + 3 \hat{j}+ 6 \hat{k} . \]
\[\text{ Let } \vec{b} \text { be a vector parallel to the vector with direction ratios }1, 2, 2 . \]
\[ \Rightarrow \vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k} \]
\[\text { Let } \theta \text{ be the angle between the the given vectors }. \]
\[\text{ Now, }\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} \]
\[ = \frac{\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right) . \left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right)}{\left| 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right|\left| \hat{i} + 2 \hat{j} + 2 \hat{k} \right|}\]
\[ = \frac{2 + 6 + 12}{\sqrt{4 + 9 + 36} \sqrt{1 + 4 + 4}}\]
\[ = \frac{20}{21} \]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{20}{21} \right)\]
APPEARS IN
संबंधित प्रश्न
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
Define direction cosines of a directed line.
What are the direction cosines of Y-axis?
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
Write the distance of the point P (x, y, z) from XOY plane.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
The distance of the point P (a, b, c) from the x-axis is
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
The angle between the two diagonals of a cube is
Find the direction cosines of a vector whose direction ratios are
1, 2, 3
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"k"`
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `vec"a", vec"b", vec"c"`
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
If the directions cosines of a line are k,k,k, then ______.
The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
