Advertisements
Advertisements
Question
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Advertisements
Solution
\[\text {The vertices of ∆ ABC are }A \left( 3, 5, - 4 \right), B \left( - 1, 1, 2 \right)\text{ and } C \left( - 5, - 5, - 2 \right) .\]
\[\text{The direction ratios of AB are} \left( - 1 - 3 \right), \left( 1 - 5 \right), \left[ 2 - \left( - 4 \right) \right], \text{i . e} . - 4, - 4, 6 . \]
\[\text{Therefore, the direction cosines of AB are}\]
\[\frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}, \frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}, \frac{6}{\sqrt{\left( - 4 \right)^2 + \left( - 4 \right)^2 + \left( 6 \right)^2}}\]
\[ = \frac{- 4}{2\sqrt{17}}, \frac{- 4}{2\sqrt{17}}, \frac{6}{2\sqrt{17}} \]
\[ = \frac{2}{\sqrt{17}}, \frac{2}{\sqrt{17}}, \frac{- 3}{\sqrt{17}}\]
\[\text{The direction ratios of BC are} \left[ - 5 - \left( - 1 \right) \right], \left( - 5 - 1 \right), \left( - 2 - 2 \right), \text{i . e} . - 4, - 6, - 4 . \]
\[\text{Therefore, the direction cosines of BC are}\]
\[\frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}, \frac{- 6}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}, \frac{- 4}{\sqrt{\left( - 4 \right)^2 + \left( - 6 \right)^2 + \left( - 4 \right)^2}}\]
\[ = \frac{- 4}{2\sqrt{17}}, \frac{- 6}{2\sqrt{17}}, \frac{- 4}{2\sqrt{17}} \]
\[ = \frac{2}{\sqrt{17}}, \frac{3}{\sqrt{7}}, \frac{2}{\sqrt{17}}\]
\[\text{The direction ratios of CA are} \left[ 3 - \left( - 5 \right) \right], \left[ 5 - \left( - 5 \right) \right], \left[ - 4 - \left( - 2 \right) \right],\text{ i . e} . 8, 10, - 2 . \]
\[\text{Therefore, the direction cosines of CA are}\]
\[\frac{8}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}, \frac{10}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}, \frac{- 2}{\sqrt{\left( 8 \right)^2 + \left( 10 \right)^2 + \left( - 2 \right)^2}}\]
\[ = \frac{8}{2\sqrt{42}}, \frac{10}{2\sqrt{42}}, \frac{- 2}{2\sqrt{42}} \]
\[ = \frac{4}{\sqrt{42}}, \frac{5}{\sqrt{42}}, \frac{- 1}{\sqrt{42}}\]
APPEARS IN
RELATED QUESTIONS
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.
(A) `0,1/sqrt2,-1/sqrt2`
(B) `0,-1/sqrt2,-1/sqrt2`
(C) `1,1/sqrt2,1/sqrt2`
(D) `0,-1/sqrt2,1/sqrt2`
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Define direction cosines of a directed line.
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
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 equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
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.
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
`hat"j"`
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 a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
If the directions cosines of a line are k,k,k, then ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
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 d.c's of a line whose direction ratios are 2, 3, –6, are ______.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
