Three Dimensional Geometry — JEE Mains Mathematics

Formulas (5)

• Direction cosines and direction ratios · Direction cosines from direction ratios
  $$(l,m,n)=\frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}},\qquad l^2+m^2+n^2=1$$
• Two lines from a pair of equations in l, m, n · Angle between the two directions
  $$\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$$
• The angle between two lines, perpendicular and parallel lines · Angle between two lines
  $$\cos\theta=\frac{|\vec d_1\cdot\vec d_2|}{|\vec d_1|\,|\vec d_2|}$$
• Writing a line: through a point, two points, or perpendicular to two lines · Parametric point and a common perpendicular direction
  $$(x_1+a\lambda,\ y_1+b\lambda,\ z_1+c\lambda),\qquad \vec d=\vec d_1\times\vec d_2$$
• Triangles and tetrahedra with coordinates · Area and volume
  $$[ABC]=\tfrac12|\vec{AB}\times\vec{AC}|,\qquad V=\tfrac16\left|[\vec{AB}\ \vec{AC}\ \vec{AD}]\right|$$

Watch out for (5)

• Ratios are not cosines until you divide
  → Direction cosines and direction ratios
• Remove a variable with the LINEAR equation first
  → Two lines from a pair of equations in l, m, n
• Coefficient of $x$, $y$, $z$ must be $+1$
  → The angle between two lines, perpendicular and parallel lines
• Step with the UNIT vector
  → Writing a line: through a point, two points, or perpendicular to two lines
• Half for a triangle, a sixth for a tetrahedron
  → Triangles and tetrahedra with coordinates

Formulas (4)

• The foot of the perpendicular on a line · Parameter of the foot
  $$t=\frac{(P-A)\cdot\vec d}{|\vec d|^2},\qquad M=A+t\vec d$$
• The image of a point in a line · Image from the foot
  $$Q=2M-P$$
• Distance of a point from a line · Distance from a line
  $$d=\frac{|\vec{AP}\times\vec d|}{|\vec d|}$$
• Triangles with a side on the line · Half-chord and area
  $$MQ=\sqrt{k^2-PM^2},\qquad [PQR]=\tfrac12\,QR\cdot PM$$

Watch out for (4)

• Divide by $|\vec d|^2$, not $|\vec d|$
  → The foot of the perpendicular on a line
• $2M-P$, not $M+P$
  → The image of a point in a line
• Divide by $|\vec d|$
  → Distance of a point from a line
• The height is $PM$, not $PQ$
  → Triangles with a side on the line

Formulas (5)

• Shortest distance between skew lines · Shortest distance between skew lines
  $$SD=\frac{\left|(\vec a_2-\vec a_1)\cdot(\vec d_1\times\vec d_2)\right|}{|\vec d_1\times\vec d_2|}$$
• Distance between parallel lines · Distance between parallel lines
  $$d=\frac{|(\vec a_2-\vec a_1)\times\vec d|}{|\vec d|}$$
• The line of shortest distance: its feet on the two lines · Conditions for the feet
  $$\vec{PQ}\cdot\vec d_1=0,\qquad \vec{PQ}\cdot\vec d_2=0$$
• Where two lines meet, and when they are coplanar · Condition for coplanar lines
  $$(\vec a_2-\vec a_1)\cdot(\vec d_1\times\vec d_2)=0$$
• A line that meets two given lines · Condition for a given direction
  $$Q-P=k\,\vec v$$

Watch out for (5)

• Absolute value, and two answers
  → Shortest distance between skew lines
• Check for parallel directions first
  → Distance between parallel lines
• Perpendicular to BOTH lines
  → The line of shortest distance: its feet on the two lines
• Always check the third equation
  → Where two lines meet, and when they are coplanar
• Two parameters, not one
  → A line that meets two given lines

Formulas (4)

• A plane from a point and a normal, three points, or its intercepts · Point-normal form
  $$a(x-x_1)+b(y-y_1)+c(z-z_1)=0$$
• The normal as a cross product: planes containing lines or perpendicular to planes · Normal from two in-plane directions
  $$\vec n=\vec u\times\vec v$$
• The family of planes through a line of intersection · Planes through a line of intersection
  $$P_1+\lambda P_2=0$$
• The angle between two planes; parallel and perpendicular planes · Angle between planes
  $$\cos\theta=\frac{|\vec n_1\cdot\vec n_2|}{|\vec n_1|\,|\vec n_2|}$$

Watch out for (4)

• Intercepts are not the normal
  → A plane from a point and a normal, three points, or its intercepts
• Perpendicular to a plane means CONTAINING its normal
  → The normal as a cross product: planes containing lines or perpendicular to planes
• Parallel to a line: dot with its DIRECTION
  → The family of planes through a line of intersection
• Take the absolute value for the acute angle
  → The angle between two planes; parallel and perpendicular planes

Formulas (3)

• Distance from a plane, parallel planes and sides of a plane · Distance of a point from a plane
  $$d=\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$
• The foot and the image of a point in a plane · Foot and image
  $$F=P-k\,\vec n,\quad Q=P-2k\,\vec n,\quad k=\frac{ax_1+by_1+cz_1+d}{a^2+b^2+c^2}$$
• Projections onto a plane · Length of a projection onto a plane
  $$\sqrt{|\vec v|^2-(\vec v\cdot\hat n)^2}$$

Watch out for (3)

• Match the normals before subtracting constants
  → Distance from a plane, parallel planes and sides of a plane
• Keep the sign of the plane's value
  → The foot and the image of a point in a plane
• Subtract the normal component, not the whole vector
  → Projections onto a plane

Formulas (3)

• Where a line meets a plane, and the ratio a plane cuts · Ratio in which a plane divides AB
  $$\frac{AC}{CB}=-\frac{P(A)}{P(B)}$$
• The angle between a line and a plane; parallel and lying in · Angle between a line and a plane
  $$\sin\theta=\frac{|\vec d\cdot\vec n|}{|\vec d|\,|\vec n|}$$
• Distance measured parallel to a given line · Distance along a direction
  $$\text{distance}=|t|\,|\vec d|,\quad P+t\vec d\ \text{on the target}$$

Watch out for (3)

• $0\cdot t=c$ means no meeting point
  → Where a line meets a plane, and the ratio a plane cuts
• SINE for a line and a plane
  → The angle between a line and a plane; parallel and lying in
• Multiply by $|\vec d|$ at the end
  → Distance measured parallel to a given line