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
Let A(3, 5,−4), B(−1, 1, 2) and C(−5, −5, −2).
Direction ratio of AB = (−1 − 3), (1 − 5), (2 − (−4))
= (−4, −4, 6)
|AB| = `sqrt((-4)^2 + (-4)^2 + (6)^2)`
= `sqrt(16 + 16 + 36)`
= `sqrt68`
= `2sqrt17`
Direction ratio of BC = (−5 − (−1), −5 − 1, −2 − 2)
= (−4, −6, −4)
|BC| = `sqrt((-4)^2 + (-6)^2 + (-4)^2)`
= `sqrt(16 + 36 + 16)`
= `sqrt68`
= `2sqrt17`
Direction ratio of CA = (−5 − 3, −5 − 5, −2 − (−4))
= (−8, −10, 2)
|CA| = `sqrt((-8)^2 + (-10)^2 + (2)^2)`
= `sqrt(64 + 100 + 4)`
= `sqrt168`
= `2sqrt42`
∴ AB are `< (-1 - 3)/|AB|, (1 - 5)/|AB|, (2 + 4)/|AB| >`
i.e., `< (-2)/sqrt17, (-2)/sqrt17, 3/sqrt17 >`
∴ d.c. of BC are `< (-5 + 1)/|BC|, (- 5 - 1)/|BC|, (- 2 -2)/|BC|>`
i.e., `< (-2)/sqrt17, (-3)/sqrt17, (-2)/sqrt17 >`
∴ d.c of CA are `< (3 + 5)/|CA|, (5 + 5)/|CA|, (- 4 + 2)/|CA|`
i.e., `< 4/sqrt42, 5/sqrt42, (-1)/sqrt42 >`
APPEARS IN
RELATED QUESTIONS
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
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`
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 a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
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
2l + 2m − n = 0, mn + ln + lm = 0
Define direction cosines of a directed line.
What are the direction cosines of Z-axis?
Write the distance of the point (3, −5, 12) from X-axis?
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 angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
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.
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
The distance of the point P (a, b, c) from the x-axis is
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
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
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
`5hat"i" - 3hat"j" - 48hat"k"`
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is
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.
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If the directions cosines of a line are k,k,k, then ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
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 ______.
The d.c's of a line whose direction ratios are 2, 3, –6, are ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
