Advertisements
Advertisements
Question
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.
Advertisements
Solution
It is given that l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines. Therefore,
Let l, m, n be the direction cosines of the line which is perpendicular to the line with direction cosines l1, m1, n1 and l2, m2, n2
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
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
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 a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
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 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
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 distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
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 distance of the point P (a, b, c) from the x-axis 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 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/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 4hat"j" + 8hat"k"`
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 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 ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
Find the direction cosine of a line which makes equal angle with coordinate axes.
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 ______.
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 ______.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
