Advertisements
Advertisements
Question
Obtain the vector equation of the line `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
Solution
The cartesian equations of the line are `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
This line is passing through the point A(– 5, – 4, – 5) and having direction ratios 3, 5, 6.
Let `bar"a"` be the position vector of the point A w.r.t. the origin and `bar"b"` be the vector parallel to the line.
Then `bar"a" = -5hat"i" - 4hat"j" - 5hat"k" and bar"b" = 3hat"i" + 5hat"j" + 6hat"k"`.
The vector equation of the line passing through `"A"(bara)` and parallel to `bar"b" "is" bar"r" = bar"a" + lambdabar"b"` where λ is a scalar.
∴ the vector equation of the required line is
`bar"r" = (-5hat"i" - 4hat"j" - 5hat"k") + lambda(3hat"i" + 5hat"j" + 6hat"k")`.
APPEARS IN
RELATED QUESTIONS
Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k" "and perpendicular to vectors" hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.
Find the vector equation of the line passing through the point having position vector `-hat"i" - hat"j" + 2hat"k" "and parallel to the line" bar"r" = (hat"i" + 2hat"j" + 3hat"k") + λ(3hat"i" + 2hat"j" + hat"k").`
Find the Cartesian equations of the line passing through A(2, 2, 1) and B(1, 3, 0).
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.
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
Find the Cartesian equation of the plane passing through A(7, 8, 6) and parallel to the XY plane.
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 plane which makes intercepts 1, 1, 1 on the co-ordinates axes.
Find the vector equation of the line which passes through the origin and the point (5, –2, 3).
Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`.
Find the Cartesian equations of the line which passes through the point (2, 1, 3) and perpendicular to the lines `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3) and x/(-3) = y/(2) = z/(5)`.
Find the vector equation of the line which passes through the origin and intersect the line x – 1 = y – 2 = z – 3 at right angle.
Find the vector equation of the line whose Cartesian equations are y = 2 and 4x – 3z + 5 = 0.
Find the coordinates of points on th line `(x - 1)/(1) = (y - 2)/(-2) = (z - 3)/(2)` which are at the distance 3 unit from the base point A(l, 2, 3).
Choose correct alternatives :
The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.
The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are ______.
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 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")`.
Find the vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`
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`.
Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form
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 and vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k"` and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `2hat"i" - hat"j" + hat"k"`
Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes
The cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(hatj + hatk)` is ______
If line joining points A and B having position vectors `6overlinea - 4overlineb + 4overlinec` and `-4overlinec` respectively, and the line joining the points C and D having position vectors `-overlinea - 2overlineb - 3overlinec` and `overlinea + 2overlineb - 5overlinec` intersect, then their point of intersection is ______
The shortest distance between A (1, 0, 2) and the line `(x + 1)/3 = (y - 2)/(-2) = (z + 1)/(-1)` is given by line joining A and B, then B in the line 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 equation of line is `(x - 1)/2 = (y + 1)/(-2) = (z + 1)/1`. The co-ordinates of the point on the line at a distance of 3 units from the point (1, -1, -1) is ______
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 ______.
Find the Cartesian equation of the plane passing through A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
