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
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`
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
Define direction cosines of a directed line.
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the distance of the point (3, −5, 12) from X-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Write the coordinates of the projection of point P (x, y, z) on XOZ-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?
Write direction cosines of a line parallel to z-axis.
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
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
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
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
1, 2, 3
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 4hat"j" + 8hat"k"`
Find the direction cosines and direction ratios for the following vector
`hat"j"`
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 the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
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 ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
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 ______.
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 ______.
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 ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
