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Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.
Concept: Distance in Lines (Point & Parallel Lines)
Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .
Concept: Distance of a Point from a Plane
The acute angle between the two planes x+y+2z = 3 and 3x -2y +2z = 7 ________.
Concept: Angle Between Two Planes
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.
Concept: Vector and Cartesian Equations of a Line
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
Concept: Vector and Cartesian Equations of a Line
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.
Concept: Vector and Cartesian Equations of a Line
Choose correct alternatives:
The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.
Concept: Vector and Cartesian Equations of a Line
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 ______.
Concept: Vector and Cartesian Equations of a Line
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
Concept: Vector and Cartesian Equations of a Line
Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.
Concept: Vector and Cartesian Equations of a Line
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)`
Concept: Vector and Cartesian Equations of a Line
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") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter
Concept: Vector and Cartesian Equations of a Line
Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2
Concept: Vector and Cartesian Equations of a Line
Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
Concept: Vector and Cartesian Equations of a Line
Find m, if the lines `(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2` and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5` are at right angles
Concept: Vector and Cartesian Equations of a Line
Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes
Concept: Vector and Cartesian Equations of a Line
Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.
Concept: Coplanarity of Two Lines
Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`
Concept: Distance in Lines (Point & Parallel Lines)
Find the vector equation of a line passing through the point `hati + 2hatj + 3hatk` and perpendicular to the vectors `hati + hatj + hatk` and `2hati - hatj + hatk`.
Concept: Vector and Cartesian Equations of a Line
Minimize `z=4x+5y ` subject to `2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0` solve using graphical method.
Concept: Methods to Solve LPP (Graphical / Corner Point Method)
