Three Dimensional Geometry Test - 1

Direction cosines of a line are the cosines of the angles made by the line with the positive direction of the coordinate axis.i.e. x- axis , y-axis and z –axis respectively.

Shortest distance between two skew lines is The line segment perpendicular to both the lines .

On comparing the given equations with :
In the cartesian form two lines


we get ;

x₁  = -1, y₁  = -1,z₁  = -1, ; a₁  = 7, b₁  = -6, c₁  = 1 and  

x₂  = 3, y₂  = 5, z₂  = 7; a₂  = 1, b₂  = -2, c₂  = 1

Now the shortest distance between the lines is given by :

By definition , The angle  θ between the planes A₁ x + B₁ y + C₁ z + D₁ = 0 and A₂ x + B₂ y + C₂ z + D₂ = 0 is given by :

The equation of the plane through the line of intersection of the planes

If l, m , n are the direction cosines of a line then , we know that,  l² + m² + n² = 1.

On comparing the given equations with: 
, we get: 

The distance of a point whose position vector is    from the plane    given by :

is a vector joining two points P(x₁ , y₁ , z₁ ) and Q(x₂ , y₂ , z₂ ). If  Direction cosines of  are given by : 

In Cartesian coordinate system Shortest distance between the lines

Find the shortest distance between the lines  

On comparing them with :

we get : 

The distance d from a point P(x₁ , y₁ , z₁ ) to the plane Ax + By + Cz + D = 0 is given by :

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are Proportional to the direction cosines of the line.

In vector form Distance between two parallel lines    given by :

If θis the acute angle between

then cosine of the angle between
these two lines is given by :


Here, 


Then, 

We have z = 2 . , it can be written as : 0x+0y+1z = 2. Compare it with lx+my+nz = d , we get ; l = 0 , m = 0 , n = 1 and d = 2 . therefore , D.C.’s of normal to the plane are 0 , 0 , 1 and distance from the origin = 2.

We have , 
2x + y + 3z –2 = 0 and x –2y + 5 = 0. Let  θ be the angle between the planes , then  

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then , the directions cosines of the line are given by :

If a line makes angles  90^∘ , 135^∘ , 45^∘ with the x, y and z –axes respectively, then the direction cosines of this line is given by :

In the vector form, equation of a plane which is at a distance d from the origin, and    is the unit vector normal to the plane through the origin is given by : 

Here , D.R ’s of normal to the plane are 1, 1 , 1 ,its D.C ‘s are :

On dividing x + y + z = 1 by √3 , we get :
 It is of the form : lx+my+nz = d , therefore , d = 1/√3 .