Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text { The given points are } A \left( 1, 2, 3 \right), B\left( 4, 5, 7 \right), C\left( - 4, 3, - 6 \right) \text{ and } D \left( 2, 9, 2 \right) . \]
\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text { and } \left( x_2 , y_2 , z_2 \right) \text { are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text { The direction ratios of AB are } \left( 4 - 1 \right), \left( 5 - 2 \right), \left( 7 - 3 \right), \text { i . e } . 3, 3, 4 . \]
\[\text { The direction ratios of CD are } \left[ 2 - \left( - 4 \right) \right], \left( 9 - 3 \right), \left[ 2 - \left( - 6 \right) \right], \text { i . e }. 6, 6, 8 . \]
\[\text { Let } \theta \text { be the angle between AB and CD } . \]
\[\text { We have } \]
\[ a_1 = 3, b_1 = 3, c_1 = 4 \]
\[ a_2 = 6, b_2 = 6, c_2 = 8\]
\[ \therefore \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}} = \frac{18 + 18 + 32}{\sqrt{9 + 9 + 16}\sqrt{36 + 36 + 64}} = \frac{68}{68} = 1\]
\[ \Rightarrow \theta = 0° \]
\[\text { Thus, the angle between AB and CD measures } 0° . \]
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; 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 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?
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
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 coordinates of the projection of point P (x, y, z) on XOZ-plane.
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.
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
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
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 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.
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
`hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
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 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 ______.
If the directions cosines of a line are k,k,k, then ______.
If a line makes angles 90°, 135°, 45° with x, y and z-axis respectively then which of the following will be its direction cosine.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
The d.c's of a line whose direction ratios are 2, 3, –6, are ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
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.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
If \[\alpha\], \[\beta\], and \[\gamma\] are the direction angles of a line, which ordered triple gives its direction cosines?
Which identity must be satisfied by the direction cosines \[(l,m,n)\] of a line?
For direction ratios \[(a,b,c)\], which set can represent the corresponding direction cosines when the direction of the line is chosen?
