Three Dimensional Coordinate Geometry II

  • Lines

A line is uniquely specified, when coordinates of 2 distinct points are known. In other words, only 1 line passes through 2 distinct points. Let P (x_1,y_1,z_1) and Q(x_2,y_2,z_2) be those 2 points.

line.jpg

 

Let R(x,y,z) be any point on that line. Then,

{\vec {PR} = t (\vec {PQ})}

since both the vectors are collinear. t is any real number (or a scalar).

In other words,

{x-x_1 = t(x_2 - x_1),\ y-y_1 = t (y_2 - y_1), \ z-z_1 = t (z_2 - z_1)}

(x_2-x_1) , (y_2-y_1) and (z_2-z_1) are known as the direction ratios of the line.

If each of the direction ratios is divided by the distance between P and Q, we get the direction cosines of the line. These are the cosines of the angles made by the line with X, Y and Z axes respectively. Let them be l,m and n.

Having studied vectors, we know that the direction cosines of a line are the components of the unit vector along that direction. Hence,

{l^2 + m^2 + n^2 = 1}

(To verify the fact that these represent the cosines of the angles, we have to use the dot product of vectors. For example, consider \hat a = l \hat i + m \hat j + n \hat k. Unit vector along X axis is \hat i. Taking the dot product \hat a \cdot \hat i we get cos (\theta_x) = l)

  • Angle between any 2 lines is given by

{cos^{-1} (l_1 l_2 + m_1 m_2 + n_1 n_2)}

  • Two lines are parallel if their direction cosines have a constant ratio. i.e.

{\frac {l_1}{l_2} = \frac {m_1}{m_2} = \frac {n_1}{n_2}}

  • 2 lines are coincident if

{l_1l_2 + m_1 m_2 + n_1 n_2 = 1}

  • 2 lines are perpendicular if

{l_1 l_2 + m_1 m_2 + n_1 n_2 = 0}

  • Projection of a segment on a line

Let P (x_1,y_1,z_1) and Q(x_2,y_2,z_2) be the endpoints of a segment and l,m,n be direction cosines of a line L. The projection of segment PQ on line L is given by

{l(x_2-x_1) + m(y_2 -y_1)+ n(z_2-z_1)}

 

  • Planes

A plane is an entity formed by 3 non-collinear points. The general equation of plane is

{ax+by+cz + d = 0}

a,b,c are the direction ratios of the vector normal to the plane. This can be easily verified by using the fact that dot product of any 2 perpendicular vectors is 0.

  • If the plane passes through (x_1,y_1,z_1) and has a normal with direction ratios a,b,c, then the equation is

{a(x-x_1)+ b(y-y_1)+c (z-z_1)= 0}

plane.jpg

 

 

  •  If A,B,C are the intercepts made by the plane with $X,Y$ and $Z$ axes respectively, then

{\frac x A + \frac y B + \frac z C = 1}

This is known as the slope-intercept form of the equation of plane.

  • If a plane is parallel to ax+by+cz + d =0, its equation is

{ax+ by+cz + d_1 = 0}

  • Angle between any 2 planes is the angle between their normals.

  •  Length of perpendicular from a point (x,y,z) on the plane ax+by+cz+d=0 is given by

{\Big| \frac {ax_1 + by_1 + cz_1 + d}{\sqrt {a^2 + b^2 + c^2}} \Big|}

  • Equation of a plane, passing through the intersection of 2 planes P_1 =0 and P_2 =0 is given by

{P_1 + \lambda P_2 =0}

\lambda is a parameter \in \mathbb R.

  • Equation of plane passing through P_1, P_2 and P_3 is given by

{\begin {vmatrix} x& y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \end {vmatrix} = 0}

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