Advertisements
Advertisements
प्रश्न
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Advertisements
उत्तर
\[\text {The vertices of ∆ ABC are }A \left( 3, 5, - 4 \right), B \left( - 1, 1, 2 \right)\text{ and } C \left( - 5, - 5, - 2 \right) .\]
\[\text{The direction ratios of AB are} \left( - 1 - 3 \right), \left( 1 - 5 \right), \left[ 2 - \left( - 4 \right) \right], \text{i . e} . - 4, - 4, 6 . \]
\[\text{Therefore, the direction cosines of AB are}\]
\[\frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}, \frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}, \frac{6}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}\]
\[ = \frac{- 4}{2\sqrt{17}}, \frac{- 4}{2\sqrt{17}}, \frac{6}{2\sqrt{17}} \]
\[ = \frac{2}{\sqrt{17}}, \frac{2}{\sqrt{17}}, \frac{- 3}{\sqrt{17}}\]
\[\text{The direction ratios of BC are} \left[ - 5 - \left( - 1 \right) \right], \left( - 5 - 1 \right), \left( - 2 - 2 \right), \text{i . e} . - 4, - 6, - 4 . \]
\[\text{Therefore, the direction cosines of BC are}\]
\[\frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}, \frac{- 6}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}, \frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}\]
\[ = \frac{- 4}{2\sqrt{17}}, \frac{- 6}{2\sqrt{17}}, \frac{- 4}{2\sqrt{17}} \]
\[ = \frac{2}{\sqrt{17}}, \frac{3}{\sqrt{7}}, \frac{2}{\sqrt{17}}\]
\[\text{The direction ratios of CA are} \left[ 3 - \left( - 5 \right) \right], \left[ 5 - \left( - 5 \right) \right], \left[ - 4 - \left( - 2 \right) \right],\text{ i . e} . 8, 10, - 2 . \]
\[\text{Therefore, the direction cosines of CA are}\]
\[\frac{8}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}, \frac{10}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}, \frac{- 2}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}\]
\[ = \frac{8}{2\sqrt{42}}, \frac{10}{2\sqrt{42}}, \frac{- 2}{2\sqrt{42}} \]
\[ = \frac{4}{\sqrt{42}}, \frac{5}{\sqrt{42}}, \frac{- 1}{\sqrt{42}}\]
APPEARS IN
संबंधित प्रश्न
Which of the following represents direction cosines of the line :
(a)`0,1/sqrt2,1/2`
(b)`0,-sqrt3/2,1/sqrt2`
(c)`0,sqrt3/2,1/2`
(d)`1/2,1/2,1/2`
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 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?
What are the direction cosines of Z-axis?
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
For every point P (x, y, z) on the xy-plane,
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Find the direction cosines of a vector whose direction ratios are
0, 0, 7
Find the direction cosines and direction ratios for the following vector
`3hat"i" + hat"j" + hat"k"`
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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
The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
If a line makes angles 90°, 135°, 45° with x, y and z-axis respectively then which of the following will be its direction cosine.
Find the direction cosine of a line which makes equal angle with coordinate axes.
What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are ______.
If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is ______.
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 ______.
