3D Geometry - 7

On solving, we get   ₁ = 0   &   µ₁ = µ₂
  = + ₁( - ) + µ₁( + )
Since   ₁ = 0
= + µ₁( + ) is required eq. of straight line.


Q.3. Prove that three lines from O with direction cosines l₁, m₁, n₁; l₂, m₂, n₂; l₃, m₃, n₃ are coplaner if
    l₁(m₂n₃ - n₂m₃) + m₁(n₂l₃ - l₂n₃) + n₁(l₂m₃ - l₃m₂) = 0

Ans: Note: Three given lines are coplanar if they have common perpendicular. Let d.c's of common be l, m, n
ll₁ + mm₁ + nn₁ = 0 ............................. (i)
    ll₂ + mm₂ + nn₂ = 0 ............................. (ii)
    ll₃ + mm₃ + nn₃ = 0 ............................. (iii)
Soplving (ii) & (iii) by cross multiplication ......

l = k(m₂n₃ - n₂m₃), m = k(n₂l₃ - n₃l₂), n = k(l₂m₃ - l₃m₂)

Substituting in (i), we get
    k(m₂n₃ - n₂m₃)l₁ + k(n₂l₃ - n₃l₂)m₁ + k(l₂m₃ - l₃m₂)n₁ = 0

l₁(m₂n₃ - n₂m₃) + m₁(n₂l₃ - n₃l₂) + n₁(l₂m₃ - l₃m₂) = = 0


Q.4. Solve the equation   x + y + z =

Ans:   x + y + z = .................................. (i)
Taking dot product by x , we get
    x.( x ) + y.( x ) + z.( x ) = .( x )
    x[] = []   x =
&  z =
 x + y + z =
[] + [] + [] = [] is required solution.


Q.5. If planes x - cy - bz = 0, cx - y + az = 0 & bx + ay - z = 0 pass through a straight line, the find value of a² + b² + c² + 2abc.

Ans:   Givem planes are:
         x - cy - bz = 0 ............................. (i)
         cx - y + az = 0 ............................. (ii)
&       bx + ay - z = 0 ............................. (iii)
Eq. of plane passing through line of intersection of plane (i) & (ii) is
    (x - cy - bz) + (cx - y + az) = 0
x(1 + c) - (c + ) + z(- b + a) = 0 ............................. (iv)
If plane (iv) & (iii) are same, then,
    
  and  

a - a³ + bc - a²bc = a²bc + ac² + ab² + bc
a(2abc + c² + b² + a² - 1) = 0
a² + b² + c² + 2abc = 1




Q.1. Show that distance of a point A() to line = + t is which is also equal to .

Ans:

Let P is point on line such that & P.V. of P is   = + t
  . = 0
    ( - ).( - ) = 0 ........................... (i)
    ( + t - ).( + t - ) = 0
    ( - + t).(t) = 0
t. - t + t² = 0
t( - ) + t²c² = 0
t = ..................................... (ii)
From (i) & (ii)
    
    
     ....................................... (iii)
Also, PA² = AB² - BP² = ||² - ||²
              
              
               ....................................... (iv)


Dumb Question: How this eq. (iv) const.

Ans: We know that   (AB)² -


Q.2. If direction cosines of variable line in two adjacent points be l, m, n & l + l, m + m, n + n show that small anhgle 5 b/w two position is
    ² = l² + m² + n²

Ans: As we know   l² + m² + n² = 1
and   (l + l)² + (m + m)² + (n + n)² = 1
l² + (l²) 2ll + m² + (m)² + 2mm + n² + (n)² + 2nn = 1
l² + m² + n² + (l)² + (m)² + (n)² + 2(ll + mm + nn) = 1
(l)² + (m)² + (n)² = - 2(ll + mm + nn) ............................ (i)
is angle b/w two positions.
    cos = l(l + l) + m(m + m) + n(n + n)
    1 - 2 sin² = 1 + ll + mm + nn ........................................... (ii)
From (i) & (ii)
    (l)² + (m)² + (n)² = 4 sin²
4()² = (l)² + (m)² + (n)²

()2 = (l)2 + (m)2 + (n)2

Dumb Question: How sin² is equal to ()² ?

Ans: Since is very small so, is very small
So,   sin as 0
So, for approsmation we have taken.


Q.3. Find image of point P(3, 5, 6) in plane 2x + y + z = 0   & find eq. of sphere having point P(3, 5, 6) & its image point of extremities of diameter.

Ans:

    2x + y + z = 0
    P(3, 5, 6)
dr's iof normal to plane (i) are 2, 1, 1
Let be image of point P in plane (i)
Eq of P R is,
     = r
  R(2r + 3, r + 5, r + 6)

[Dumb Question: How eq. of line PR come ?
  Ans: d.r's of normal to plane is same as d.r's of line PR i.e. 2, 1, 1
         So, we get eq. of line PR]

But R lies on (i)
  2(2r + 3) +(r + 5) + (r + 6) = 0
    6r + 17 = 0   r = -
    R
Since R lies mid point of PQ
    
    

    Q =
Eq. of sphere when extremities (x₁, y₁, z₁) & (x₂, y₂, z₂) are given:
    (x - x₁)(x - x₂) + (y - y₁)(y - y₂) + (z - z₁)(z - z₂) = 0
    (x - 3) + (y - 5) + (z - 6) = 0
x² + y² + z² + = 0
x² + y² + z² + = 0
3x² + 3y² + 3z² + 16x - 13y - 19z - 79 = 0


Key words

Direction COsines.
Direction Ratios.
Reflection of a point.
Skew Lines.
Shortest distance.
Intersection of lines.
Normal. FV
Angle bisector.
Tangency of plane to sphere.