Advertisements
Advertisements
प्रश्न
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Advertisements
उत्तर
\[\text { We know that the direction ratios of the line passing through the points } \left( x_1 , y_1 , z_1 \right) \text { and } \left( x_2 , y_2 , z_2 \right) \text { are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text{ Let the first two points be A } \left( 4, 7, 8 \right) \text{ and } B \left( 2, 3, 4 \right) . \]
\[\text{ Thus, the direction ratios of AB are } \left( 2 - 4 \right), \left( 3 - 7 \right), \left( 4 - 8 \right), \text{ i . e } . - 2, - 4, - 4 . \]
\[\text{ Similarly, let the other two points be C } \left( - 1, - 2, 1 \right) \text{ and } D\left( 1, 2, 5 \right) . \]
\[\text{ Thus, the direction ratios of CD are } \left[ 1 - \left( - 1 \right) \right], \left[ 2 - \left( - 2 \right) \right], \left( 5 - 1 \right),\text{ i . e} . 2, 4, 4 . \]
\[\text{ It can be seen that the direction ratios of CD are - 1 times that of AB, i . e . they are proportional }. \]
\[\text { Therefore, AB and CD are parallel lines } .\]
APPEARS IN
संबंधित प्रश्न
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −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.
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
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.
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 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
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Find the distance of the point (2, 3, 4) from the x-axis.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
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.
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 angle between the two diagonals of a cube is
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines of a vector whose direction ratios are
1, 2, 3
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"`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
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
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 ______.
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
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.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector 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 ______.
Any three numbers proportional to the direction cosines of a line are called its:
Which proportion correctly expresses the relationship between direction ratios \[(a,b,c)\] and direction cosines \[(l,m,n)\]?
