Advertisements
Advertisements
प्रश्न
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).
Advertisements
उत्तर
\[\text { We know that two lines with direction ratios } a_1 , b_1 , c_1 \text { and } a_2 , b_2 , c_2 \text { are perpendicular if } a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 . \]
\[\text { The direction ratios of the line passing through the points }\left( 1, - 1, 2 \right) \text{ and } \left( 3, 4, - 2 \right) \text{ are } \left( 3 - 1 \right), \left[ 4 - \left( - 1 \right) \right], \left( - 2 - 2 \right), \text { i . e } . 2, 5, - 4 . \]
\[ \Rightarrow a_1 = 2, b_1 = 5, c_1 = - 4\]
\[\text { Similarly, the direction ratios of the line passing through the points } \left( 0, 3, 2 \right) \text { and } \left( 3, 5, 6 \right) \text { are }\left( 3 - 0 \right), \left( 5 - 3 \right), \left( 6 - 2 \right), \text{ i . e} . 3, 2, 4 . \]
\[ \Rightarrow a_2 = 3, b_2 = 2, c_2 = 4\]
\[ \therefore a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 \times 3 + 5 \times 2 + \left( - 4 \right) \times 4 = 6 + 10 - 16 = 0\]
` \text{ Thus, 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) } `
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
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.
Find the direction cosines of a line which makes equal angles with the coordinate axes.
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) .
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
Define direction cosines of a directed line.
What are the direction cosines of X-axis?
What are the direction cosines of Y-axis?
Write the distance of the point (3, −5, 12) from X-axis?
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
For every point P (x, y, z) on the xy-plane,
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
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
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
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"`
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"k"`
A line makes equal angles with co-ordinate axis. Direction cosines of this 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 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 line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis 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 the directions cosines of a line are k,k,k, then ______.
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 ______.
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 ______.
A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval ______.
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 foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
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 ______.
