Formula sheet: Three Dimensional Geometry

Three Dimensional Geometry15 min readLesson 1 of 7

1.1 How this sheet is arranged Everything in this chapter is about lines in space. The sheet is grouped by the four jobs a line question can ask: report a direction, write the equation, find an angle, and find a distance. The first step in most problems is to read off a point and a direction vector from the given equation, or to build the direction vector by subtracting two points. 1.2 Direction cosines and direction ratios If a line makes angles alpha, beta and gamma with the positive x, y and z axes, the cosines of these angles are its direction cosines. Any three numbers proportional to them are direction ratios.

l = cos α, m = cos β, n = cos γ

Direction cosines of a directed line, written l, m, n

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

Direction cosines always satisfy this identity

l = \pm \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \pm \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \pm \frac{c}{a ^{2} + b ^{2} + c ^{2}}

From direction ratios a, b, c to direction cosines, with one common sign

1.3 Equations of a line A line in space is fixed by one point on it and one direction along it. The vector form keeps a position vector and a direction vector; the cartesian form is the same statement written as ratios.

r = a + λ b

Vector equation of the line through point a with direction b; lambda is a real parameter

\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}

Cartesian equation of the line through (x1, y1, z1) with direction ratios a, b, c

r = a + λ (b - a)

Vector equation of the line through points with position vectors a and b

\frac{x - x _{1}}{x _{2} - x _{1}} = \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{z - z _{1}}{z _{2} - z _{1}}

Cartesian form of the line through two given points

1.4 Angle between two lines The angle between two lines is the angle between their direction vectors. The modulus keeps the reported angle acute, and the cosine formula is the dot product in disguise.

cos θ = \frac{b _{1} \cdot b _{2}}{∣ b _{1} ∣∣ b _{2} ∣}

Angle between lines with direction vectors b1 and b2

b_{1} \cdot b_{2} = 0 ⟺ lines are perpendicular

Perpendicular test in vector or direction-ratio form

b_{1} \times b_{2} = 0 ⟺ lines are parallel

Parallel test; equivalently the direction ratios are proportional

1.5 Shortest distance between two lines Two lines in space may meet, be parallel, or be skew. The shortest distance is zero for intersecting lines, is found from a projection for skew lines, and from a cross product for parallel lines.

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

Shortest distance between skew lines r = a1 + lambda b1 and r = a2 + mu b2

d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}

Shortest distance between parallel lines with common direction b

(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0 ⟺ the lines intersect or are coplanar

Zero triple product means the two lines lie in one plane; it is also the intersecting test

1.6 Reading the table Question asks Start with Direction of a line direction ratios from denominators, then direction cosines Equation from a point and a direction vector form, then convert to cartesian Equation from two points direction is the difference of position vectors Angle between lines direction vectors into the cosine formula, take modulus Lines perpendicular or parallel dot product zero, or direction ratios proportional Are the lines coplanar triple product of the joining vector and the two directions Shortest distance triple product divided by the cross product magnitude Distance between parallel lines difference of points crossed with the direction, divided by its length 1.7 Common mistakes to avoid A zero denominator in the cartesian form is allowed: it records that the corresponding direction ratio is zero, and the ratio is not evaluated but read as a condition. Direction ratios may be multiplied by any nonzero scalar, but direction cosines are fixed up to one common sign. In the angle formula the modulus is needed only when the question asks for the acute angle; for the angle between two directed lines the sign carries information. In the skew-distance formula the numerator is a scalar triple product, so the mod is outside the whole expression, not inside a single factor. Finally, if the shortest distance comes out as zero, the lines intersect, and the two-line problem reduces to finding the point of intersection.

Question asks	Start with
Direction of a line	direction ratios from denominators, then direction cosines
Equation from a point and a direction	vector form, then convert to cartesian
Equation from two points	direction is the difference of position vectors
Angle between lines	direction vectors into the cosine formula, take modulus
Lines perpendicular or parallel	dot product zero, or direction ratios proportional
Are the lines coplanar	triple product of the joining vector and the two directions
Shortest distance	triple product divided by the cross product magnitude
Distance between parallel lines	difference of points crossed with the direction, divided by its length

in hinglish

Formula sheet Three Dimensional Geometry ka poora naksha hai. Direction cosines l, m, n = cos α, cos β, cos γ hote hain aur l²+m²+n² = 1; direction ratios se cosines nikalne ke liye \sqrt(a²+b²+c²) se divide karo, sign common rakho. Line ka vector form r = a + λb hai; cartesian form mein numerators se point aur denominators se direction ratios milte hain, aur do points se direction b - a hoti hai. Angle ke liye cosθ = |b1\cdotb2|/(|b1||b2|); perpendicular par dot zero aur parallel par cross zero ya ratios proportional. Shortest distance skew lines ke liye |(a2-a1)\cdot(b1\timesb2)|/|b1\timesb2| aur parallel lines ke liye |(a2-a1)\timesb|/|b|. Triple product zero ho to lines coplanar ya intersecting hain. Exam mein pehle point aur direction nikalo, phir kaam ke hisaab se formula chuno, aur zero denominator ko ratio samjho, divide mat karo.