Advertisements
Advertisements
Question
Find the vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`
Advertisements
Solution
The given equation of a line passes through the point A(0, 1, 2) and the direction ratios of the line are 1, 2, 3.
Let `bar"a"` be the position vector of point A.
Let `bar"b"` be the vector parallel to this line.
∴ `bar"a" = hat"j" + 2hat"k"` and `bar"b" = hat"i" + 2hat"j" + 3hat"k"`
The vector equation of a line passing through a point with position vector `bar"a"` and parallel to `bar"b"` is `bar"r" = bar"a" + lambdabar"b"`.
The vector equation of the given line is `bar"r" = (hat"j" + 2hat"k") + lambda(hat"i" + 2hat"j" + 3hat"k")`
APPEARS IN
RELATED QUESTIONS
Find the vector equation of the line passing through the point having position vector `-2hat"i" + hat"j" + hat"k" "and parallel to vector" 4hat"i" - hat"j" + 2hat"k"`.
Find the vector equation of the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.
A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
Find the Cartesian equation of the plane passing through A( -1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
The foot of the perpendicular drawn from the origin to a plane is M(1,0,0). Find the vector equation of the plane.
Find the vector equation of the line which passes through the point (3, 2, 1) and is parallel to the vector `2hat"i" + 2hat"j" - 3hat"k"`.
Find the Cartesian equations of the line which passes through the point (–2, 4, –5) and parallel to the line `(x + 2)/(3) = (y - 3)/(5) = (z + 5)/(6)`.
Find the vector and Cartesian equations of the line passing through the point (–1, –1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z − 2.
Solve the following :
Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hat"i" + hat"j" + 2hat"k"`.
Find the vector equation of the plane passing through the points A(1, -2, 1), B(2, -1, -3) and C(0, 1, 5).
Solve the following :
Find the cartesian equation of the plane passing through A(1,-2, 3) and direction ratios of whose normal are 0, 2, 0.
Solve the following :
Find the cartesian equation of the plane passing through A(7, 8, 6) and parallel to the plane `bar"r".(6hat"i" + 8hat"j" + 7hat"k")` = 0.
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
Solve the following :
A plane makes non zero intercepts a, b, c on the coordinate axes. Show that the vector equation of the plane is `bar"r".(bchat"i" + cahat"j" + abhat"k")` = abc.
Solve the following :
Find the vector equation of the plane passing through the point A(– 2, 3, 5) and parallel to the vectors `4hat"i" + 3hat"k" and hat"i" + hat"j"`.
Solve the following :
Find the cartesian equation of the plane `bar"r" = lambda(hat"i" + hat"j" - hat"k") + mu(hat"i" + 2hat"j" + 3hat"k")`.
Solve the following :
Find the vector equation of the plane passing through the origin and containing the line `bar"r" = (hat"i" + 4hat"j" + hat"k") + lambda(hat"i" + 2hat"j" + hat"k")`.
Solve the following :
Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, –2) at right angle.
Solve the following :
Show that the lines x = y, z = 0 and x + y = 0, z = 0 intersect each other. Find the vector equation of the plane determined by them.
Find the Cartesian equations of the line passing through A(3, 2, 1) and B(1, 3, 1).
Verify if the point having position vector `4hat"i" - 11hat"j" + 2hat"k"` lies on the line `bar"r" = (6hat"i" - 4hat"j" + 5hat"k") + lambda (2hat"i" + 7hat"j" + 3hat"k")`
Find the Cartesian equation of the line passing through A(1, 2, 3) and having direction ratios 2, 3, 7
Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.
Find the direction ratios of the line perpendicular to the lines
`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`
Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form
Find Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `hat"i" + 2hat"j" - hat"k"`
Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2
Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes
The cartesian coordinates of the point on the parabola y2 = x whose parameter is ____________.
The vector equation of the line passing through `4hati - hatj + 2hatk` and parallel to `-2hati - hatj + hatk` is ______
The cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(hatj + hatk)` is ______
If the line passes through the points P(6, -1, 2), Q(8, -7, 2λ) and R(5, 2, 4) then value of λ is ______.
The lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if ______
The line passing through the points (5, 1, a) and (3, b, 1) crosses the YZ – plane at the point `(0, 17/2, (-13)/2)`, then ______.
A line passes through the point of intersection of the lines 3x + y + 1 = 0 and 2x – y + 3 = 0 and makes equal intercepts with axes. The equation of the line is ______.
Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
Find the vector equation of the line passing through the points A(2, 3, –1) and B(5, 1, 2).
Show that the lines `(x - 1)/1 = (y - 2)/2 = (z + 1)/-1` and `x/2 = (y - 3)/2 = z/(-1)` do not intersect.
