Advertisements
Advertisements
Question
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).
Advertisements
Solution
\[\text { We know that two lines with direction ratios } a_1 , b_1 , c_1 \text { and } a_2 , b_2 , c_2 \text { are perpendicular if } a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 . \]
\[\text { The direction ratios of the line passing through the points }\left( 1, - 1, 2 \right) \text{ and } \left( 3, 4, - 2 \right) \text{ are } \left( 3 - 1 \right), \left[ 4 - \left( - 1 \right) \right], \left( - 2 - 2 \right), \text { i . e } . 2, 5, - 4 . \]
\[ \Rightarrow a_1 = 2, b_1 = 5, c_1 = - 4\]
\[\text { Similarly, the direction ratios of the line passing through the points } \left( 0, 3, 2 \right) \text { and } \left( 3, 5, 6 \right) \text { are }\left( 3 - 0 \right), \left( 5 - 3 \right), \left( 6 - 2 \right), \text{ i . e} . 3, 2, 4 . \]
\[ \Rightarrow a_2 = 3, b_2 = 2, c_2 = 4\]
\[ \therefore a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 \times 3 + 5 \times 2 + \left( - 4 \right) \times 4 = 6 + 10 - 16 = 0\]
` \text{ Thus, 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) } `
APPEARS IN
RELATED QUESTIONS
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`
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
Find the angle between the lines whose direction cosines are given by the equations
2l + 2m − n = 0, mn + ln + lm = 0
What are the direction cosines of Z-axis?
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
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
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
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
`3hat"i" - 4hat"j" + 8hat"k"`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
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 `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"`
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
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 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 line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
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.
What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.
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 ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
Which identity must be satisfied by the direction cosines \[(l,m,n)\] of a line?
In the relation between direction ratios and direction cosines, what does the sign \[\pm\] in \[l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}}\] depend on?
For the line directed from \[P(x_1,y_1,z_1)\] to \[Q(x_2,y_2,z_2)\], which expression is the direction cosine \[m\]?
