1. Direction Cosines and Direction Ratios

  • Direction Cosines (DCs): If a directed line makes angles α,β,γ\alpha, \beta, \gamma with the positive x, y, and z-axes respectively, then its direction cosines are l=cos⁡α,m=cos⁡β,n=cos⁡γ.l = \cos\alpha, \qquad m = \cos\beta, \qquad n = \cos\gamma. These describe the orientation of the line in space.
  • Fundamental Identity: Every set of direction cosines satisfies l2+m2+n2=1.l^2 + m^2 + n^2 = 1. This is one of the most important identities in 3D Geometry and is used repeatedly to find unknown angles or unknown direction cosines.
  • Direction Ratios (DRs): Any three numbers a,b,ca, b, c proportional to the direction cosines of a line are called direction ratios of that line. They give the same direction, but they are not required to satisfy any unit-length condition.
  • Relation between DCs and DRs: If a,b,ca, b, c are direction ratios, then the corresponding direction cosines are l=±aa2+b2+c2,m=±ba2+b2+c2,n=±ca2+b2+c2.l = \pm \frac{a}{\sqrt{a^2+b^2+c^2}}, \qquad m = \pm \frac{b}{\sqrt{a^2+b^2+c^2}}, \qquad n = \pm \frac{c}{\sqrt{a^2+b^2+c^2}}. The sign depends on the orientation chosen for the directed line.
  • Line Joining Two Points: For points P(x1,y1,z1)P(x_1, y_1, z_1) and Q(x2,y2,z2)Q(x_2, y_2, z_2), the direction ratios of the line PQPQ are (x2−x1),(y2−y1),(z2−z1).(x_2-x_1), \quad (y_2-y_1), \quad (z_2-z_1). After this, divide by the magnitude to obtain the direction cosines if required.

2. Equation of a Line

  • Line through a Point and Parallel to a Given Vector: Suppose the line passes through the point with position vector a⃗\vec{a} and is parallel to the vector b⃗\vec{b}.
  • Vector Form: r⃗=a⃗+λb⃗,\vec{r} = \vec{a} + \lambda\vec{b}, where λ\lambda is a real parameter.
  • Cartesian Form: If the point is (x1,y1,z1)(x_1,y_1,z_1) and the direction ratios are a,b,ca,b,c, then x−x1a=y−y1b=z−z1c.\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}.
  • Line through Two Given Points: If the line passes through points (x1,y1,z1)(x_1,y_1,z_1) and (x2,y2,z2)(x_2,y_2,z_2), then its direction vector is obtained by subtracting coordinates.
  • Vector Form: r⃗=a⃗+λ(b⃗−a⃗),\vec{r} = \vec{a} + \lambda(\vec{b}-\vec{a}), where a⃗\vec{a} and b⃗\vec{b} are the position vectors of the two points.
  • Cartesian Form: x−x1x2−x1=y−y1y2−y1=z−z1z2−z1.\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}.
  • Special Case: If one of the direction ratios is zero, then the corresponding coordinate remains constant. For example, if the DRs are (0,0,1)(0,0,1), then the line is parallel to the z-axis.

3. Angles and Shortest Distance for Lines

  • Angle Between Two Lines: If b⃗1\vec{b}_1 and b⃗2\vec{b}_2 are direction vectors of two lines, then the acute angle θ\theta between the lines is cos⁡θ=∣b⃗1⋅b⃗2∣b⃗1∣ ∣b⃗2∣∣.\cos\theta = \left| \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1|\,|\vec{b}_2|} \right|. The absolute value is taken because the angle between two lines is generally taken to be the acute angle.
  • Shortest Distance Between Two Skew Lines: If the lines are non-parallel and non-intersecting, then they are skew. Their shortest distance is d=∣(a⃗2−a⃗1)⋅(b⃗1×b⃗2)∣b⃗1×b⃗2∣∣.d = \left| \frac{(\vec{a}_2-\vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right|. Here, (b⃗1×b⃗2)(\vec{b}_1 \times \vec{b}_2) gives a vector perpendicular to both lines.
  • Important Note: If the two lines are non-parallel and the scalar triple product in the numerator becomes zero, then the shortest distance is zero, so the lines intersect and are coplanar.
  • Shortest Distance Between Parallel Lines: If the lines are parallel with common direction vector b⃗\vec{b}, then d=∣(a⃗2−a⃗1)×b⃗∣∣b⃗∣.d = \frac{|(\vec{a}_2-\vec{a}_1) \times \vec{b}|}{|\vec{b}|}. This formula is different from the skew-line formula and must be used only when the lines are parallel.

4. Equation of a Plane

  • Normal Form: If n^\hat{n} is a unit normal vector to a plane and dd is the perpendicular distance of the plane from the origin, then r⃗⋅n^=d.\vec{r} \cdot \hat{n} = d. If the direction cosines of the unit normal are l,m,nl,m,n, then the Cartesian form is lx+my+nz=d.lx + my + nz = d.
  • Plane through a Point with Given Normal Vector: If the plane passes through the point with position vector a⃗\vec{a} and has normal vector N⃗\vec{N}, then
  • Vector Form: (r⃗−a⃗)⋅N⃗=0.(\vec{r} - \vec{a}) \cdot \vec{N} = 0.
  • Cartesian Form: If the point is (x1,y1,z1)(x_1,y_1,z_1) and the normal has components (A,B,C)(A,B,C), then A(x−x1)+B(y−y1)+C(z−z1)=0.A(x-x_1) + B(y-y_1) + C(z-z_1) = 0.
  • Plane through Three Non-Collinear Points: If a plane passes through three non-collinear points with position vectors a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c}, then (r⃗−a⃗)⋅[(b⃗−a⃗)×(c⃗−a⃗)]=0.(\vec{r}-\vec{a}) \cdot [ (\vec{b}-\vec{a}) \times (\vec{c}-\vec{a}) ] = 0. This works because the cross product gives a normal to the plane.
  • Intercept Form of a Plane: If a plane cuts intercepts a,b,ca,b,c on the x, y, z-axes respectively, then xa+yb+zc=1.\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1.
  • Family of Planes Through the Intersection of Two Intersecting Planes: If two intersecting planes are π1=0\pi_1=0 and π2=0\pi_2=0, then every plane through their line of intersection is given by π1+λπ2=0.\pi_1 + \lambda \pi_2 = 0. This formula is extremely important in competitive exams.

5. Coplanarity, Angles, and Distance to a Plane

  • Coplanarity of Two Lines: Two lines with vector equations r⃗=a⃗1+λb⃗1\vec{r}=\vec{a}_1+\lambda\vec{b}_1 and r⃗=a⃗2+μb⃗2\vec{r}=\vec{a}_2+\mu\vec{b}_2 are coplanar if and only if (a⃗2−a⃗1)⋅(b⃗1×b⃗2)=0.(\vec{a}_2-\vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0. This means the scalar triple product is zero, so the three vectors lie in one plane.
  • Angle Between Two Planes: If n⃗1\vec{n}_1 and n⃗2\vec{n}_2 are normal vectors to the two planes, then the acute angle θ\theta between the planes is cos⁡θ=∣n⃗1⋅n⃗2∣n⃗1∣ ∣n⃗2∣∣.\cos\theta = \left| \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1|\,|\vec{n}_2|} \right|.
  • Angle Between a Line and a Plane: If b⃗\vec{b} is the direction vector of the line and n⃗\vec{n} is the normal vector of the plane, then the angle ϕ\phi between the line and the plane is sin⁡ϕ=∣b⃗⋅n⃗∣b⃗∣ ∣n⃗∣∣.\sin\phi = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}|\,|\vec{n}|} \right|. This is because the angle between a line and a plane is complementary to the angle between the line and the plane's normal.
  • Distance of a Point from a Plane: For the plane Ax+By+Cz+D=0,Ax + By + Cz + D = 0, the perpendicular distance of the point (x1,y1,z1)(x_1,y_1,z_1) is d=∣Ax1+By1+Cz1+D∣A2+B2+C2.d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}.
  • Distance Between Two Parallel Planes: If the planes are Ax+By+Cz+D1=0andAx+By+Cz+D2=0,Ax + By + Cz + D_1 = 0 \quad \text{and} \quad Ax + By + Cz + D_2 = 0, then the distance between them is d=∣D1−D2∣A2+B2+C2.d = \frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}. Before using this formula, make sure the coefficients of x,y,zx,y,z are exactly the same in both equations.

Important Exam Tips for Board Exams

  • Always Standardize Cartesian Forms: Before reading a point or direction ratios from a line equation, first rewrite it in the strict standard form x−x1a=y−y1b=z−z1c.\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}. For example, rewrite 2−y3\frac{2-y}{3} as y−2−3\frac{y-2}{-3}. This avoids sign mistakes.
  • Write Every Step in Distance Problems: In shortest-distance questions, first compute the cross product, then its magnitude, then the scalar triple product, and only after that substitute into the formula. This presentation is important in board exams.
  • Use the General Point Method Frequently: If a line is given in symmetric form, write x−x1a=y−y1b=z−z1c=λ,\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda, so that a general point on the line becomes (x1+aλ, y1+bλ, z1+cλ).(x_1+a\lambda,\ y_1+b\lambda,\ z_1+c\lambda). This method is the safest for finding intersection points, feet of perpendiculars, and reflected images.
  • Do Not Confuse Point and Direction: In 3D geometry, a point such as (1,2,3)(1,2,3) and a direction vector such as (1,2,3)(1,2,3) may look similar numerically, but geometrically they mean different things. Always state clearly whether you are using a point or a direction vector.

Important Exam Tips for JEE Main & Advanced

  • Use the Family of Planes Shortcut Immediately: If a plane has to pass through the intersection of two planes, do not first find the line of intersection unless absolutely necessary. Start directly with π1+λπ2=0.\pi_1 + \lambda\pi_2 = 0. Then use the extra condition to get λ\lambda.
  • Memorize the Image and Foot Formula: For the plane Ax+By+Cz+D=0Ax+By+Cz+D=0 and point (x1,y1,z1)(x_1,y_1,z_1), x′−x1A=y′−y1B=z′−z1C=−2(Ax1+By1+Cz1+D)A2+B2+C2\frac{x' - x_1}{A} = \frac{y' - y_1}{B} = \frac{z' - z_1}{C} = \frac{-2(Ax_1+By_1+Cz_1+D)}{A^2+B^2+C^2} gives the image of the point in the plane. Replacing −2-2 by −1-1 gives the foot of the perpendicular.
  • Switch to Vectors When Geometry Looks Messy: Many 3D problems become shorter in vector language. Coplanarity becomes scalar triple product zero, perpendicularity becomes dot product zero, and normals often come from cross products.
  • Choose Smart Coordinates or Parameters: In JEE Main, saving time is critical. When solving for a point on the line of intersection of two planes, set one variable conveniently, often z=0z=0, and solve the resulting two linear equations. This is much faster than full elimination in many questions.