Advertisements
Advertisements
प्रश्न
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Advertisements
उत्तर
\[\text{ Suppose the points are A } \left( 2, 3, 4 \right), B \left( - 1 . - 2, 1 \right) \text { and } C \left( 5, 8, 7 \right) . \]
\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text{ and } \left( x_2 , y_2 , z_2 \right) \text{ are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text{ The direction ratios of AB are } \left( - 1 - 2 \right), \left( - 2 - 3 \right), \left( 1 - 4 \right), \text{ i . e }. - 3, - 5, - 3 . \]
\[\text{ The direction ratios of BC are } \left( 5 - \left( - 1 \right) \right), \left( 8 - \left( - 2 \right) \right), \left( 7 - 1 \right), \text { i . e } . 6, 10, 6 . \]
\[ \text{ It can be seen that the direction ratios of BC are - 2 times that of AB, i . e . they are proportional . Therefore, AB is parallel to BC }. \]
\[\text { Since point B is common in both AB and BC, points A, B, and C are collinear } .\]
APPEARS IN
संबंधित प्रश्न
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
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
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.
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 − m + 2n = 0 and mn + nl + lm = 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 X-axis?
What are the direction cosines of Z-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write direction cosines of a line parallel to z-axis.
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
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
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
The angle between the two diagonals of a cube 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 and direction ratios for the following vector
`hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
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.
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.
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
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 ______.
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.
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 ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
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 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 ______.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
