Advertisements
Advertisements
प्रश्न
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
उत्तर
Let θ be the acute angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Then,
`costheta=|(a_1a_2+b_1b_2+c_1c_2)/(sqrt(a_1^2+b_1^2+c_1^2).sqrt(a_2^2+b_2^2+c_2^2))|`
`cos theta=|(4(3)+(-3)(4)+5(5))/(sqrt(4^2+(3)^2+5^2).sqrt(3^2+4^2+5^2))|`
`=|(12-12+25)/(sqrt(16+9+25).sqrt(9+16+25))|`
`=|25/50|=1/2`
`costheta=1/2`
`theta=cos^-1(1/2)=pi/3`
The angle between the lines is `pi/3`
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
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.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
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
If a line has direction ratios 2, −1, −2, determine 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.
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.
Define direction cosines of a directed line.
What are the direction cosines of X-axis?
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.
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.
The angle between the two diagonals of a cube is
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"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 `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
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
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
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 ______.
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 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 ______.
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 ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
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 ______.
