Advertisements
Advertisements
Question
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
Solution
\[\text { The given points are } A \left( 1, 2, 3 \right), B\left( 4, 5, 7 \right), C\left( - 4, 3, - 6 \right) \text{ and } D \left( 2, 9, 2 \right) . \]
\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text { and } \left( x_2 , y_2 , z_2 \right) \text { are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text { The direction ratios of AB are } \left( 4 - 1 \right), \left( 5 - 2 \right), \left( 7 - 3 \right), \text { i . e } . 3, 3, 4 . \]
\[\text { The direction ratios of CD are } \left[ 2 - \left( - 4 \right) \right], \left( 9 - 3 \right), \left[ 2 - \left( - 6 \right) \right], \text { i . e }. 6, 6, 8 . \]
\[\text { Let } \theta \text { be the angle between AB and CD } . \]
\[\text { We have } \]
\[ a_1 = 3, b_1 = 3, c_1 = 4 \]
\[ a_2 = 6, b_2 = 6, c_2 = 8\]
\[ \therefore \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}} = \frac{18 + 18 + 32}{\sqrt{9 + 9 + 16}\sqrt{36 + 36 + 64}} = \frac{68}{68} = 1\]
\[ \Rightarrow \theta = 0° \]
\[\text { Thus, the angle between AB and CD measures } 0° . \]
APPEARS IN
RELATED QUESTIONS
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 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.
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Define direction cosines of a directed line.
What are the direction cosines of Y-axis?
Write the distance of the point (3, −5, 12) from 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 distance of the point P (x, y, z) from XOY plane.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
For every point P (x, y, z) on the xy-plane,
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 P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
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 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`
Verify whether the following ratios are direction cosines of some vector or not
`4/3, 0, 3/4`
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"k"`
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `vec"a", vec"b", vec"c"`
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
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 a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ 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 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 d.c's of a line whose direction ratios are 2, 3, –6, are ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
