Formula sheet: Vector Algebra

Vector Algebra16 min readLesson 1 of 7

1.1 How this sheet is arranged The formulas below are grouped by the job they do, not by the order in the book. Read the heading, then the formula, then the line that says when to reach for it. In a vector problem the first step is almost always to write each vector in component form, because every formula on this sheet becomes arithmetic once the components are known. 1.2 Magnitude, unit vector and direction The length of a vector is found from its components. A unit vector keeps the direction and drops the length, which is how direction is reported without units.

∣ a ∣ = x^{2} + y^{2} + z^{2}

The magnitude of a vector a = x i + y j + z k

a^= \frac{a}{∣ a ∣}

Unit vector in the direction of a, for a not equal to the zero vector

l = cos α, m = cos β, n = cos γ

Direction cosines of a vector that makes angles alpha, beta, gamma with the axes

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

Direction cosines always satisfy this identity; direction ratios x, y, z need not

The direction ratios of a vector are its components taken in order, x : y : z. If a line has direction ratios a, b, c, its direction cosines are those ratios divided by the square root of the sum of their squares, with one common sign chosen. 1.3 Points, joining vectors and section formula For points A and B with position vectors a and b, the vector from A to B points from the first point to the second. The section formula splits the segment in a given ratio.

A B = b - a

Vector joining A to B, where a and b are position vectors

r = \frac{m b + n a}{m + n}

Position vector of the point dividing AB internally in the ratio m : n

r = \frac{m b - n a}{m - n}

Position vector of the point dividing AB externally in the ratio m : n

r = \frac{a + b}{2}

Midpoint of the segment joining A and B

1.4 Dot product, angle and projection The dot product answers two questions: how much do two vectors point the same way, and how much of one lies along the other.

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Scalar product from lengths and angle, and from components

cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}

Angle between two nonzero vectors

projection of a on b = \frac{a \cdot b}{∣ b ∣}

Scalar projection; it is negative when the angle is obtuse

a ⊥ b ⟺ a \cdot b = 0

Perpendicularity test for nonzero vectors

∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2 a \cdot b

Used when a length of a sum or difference is given, or a dot product is hidden in a length

∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣

Triangle inequality; equality holds only when the vectors are parallel and point the same way

1.5 Cross product and areas The cross product answers a different pair of questions: which direction is perpendicular to both vectors, and how much area do they span.

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

Magnitude of the vector product from lengths and angle

a \times b = i^a_{1} b_{1} j^a_{2} b_{2} k^a_{3} b_{3}

Component form, expanded along the first row

a ∥ b ⟺ a \times b = 0

Parallel or collinear test; the zero vector appears when the sine of the angle is zero

∣ a \times b ∣^{2} + (a \cdot b)^{2} = ∣ a ∣^{2} ∣ b ∣^{2}

Area from the dot product as well; useful when the lengths and the dot product are known and the area is asked

area of a triangle = \frac{1}{2} ∣ a \times b ∣, area of a parallelogram = ∣ a \times b ∣

Where a and b are the two sides from a common vertex

n^= \pm \frac{a \times b}{∣ a \times b ∣}

Unit vector perpendicular to the plane of a and b; both signs are correct answers

1.6 Standard products to memorise The unit vectors along the axes multiply in a cycle. For the dot product, i.i = j.j = k.k = 1 and every mixed pair gives zero. For the cross product, i x j = k, j x k = i and k x i = j, while reversing any pair flips the sign. Task in the question Formula to start with Find a length magnitude from components Find direction only unit vector, or direction cosines Prove two vectors perpendicular dot product equals zero Find angle between vectors cosine formula with dot product Find component along a direction scalar projection Prove points collinear one joining vector is a scalar multiple of another Find area of triangle or parallelogram half or full magnitude of the cross product Find a perpendicular direction cross product, then divide by its magnitude 1.7 Common mistakes to avoid A negative scalar projection is a real answer, not an error; only a length is always positive. The direction cosines satisfy l squared plus m squared plus n squared equals one, but direction ratios do not have to. When a question asks for the angle between two vectors, the answer lies between 0 and pi, so the absolute value of the cosine is not enough: keep the sign to decide obtuse or acute. In the cross product the order matters, because reversing the order reverses the direction. Area questions always want a positive number, so take the modulus before halving.

Task in the question	Formula to start with
Find a length	magnitude from components
Find direction only	unit vector, or direction cosines
Prove two vectors perpendicular	dot product equals zero
Find angle between vectors	cosine formula with dot product
Find component along a direction	scalar projection
Prove points collinear	one joining vector is a scalar multiple of another
Find area of triangle or parallelogram	half or full magnitude of the cross product
Find a perpendicular direction	cross product, then divide by its magnitude

in hinglish

Is formula sheet Vector Algebra ka poora naksha hai. Sabse pehle har vector ko component form mein likho: a = x i + y j + z k, phir |a| = \sqrt(x²+y²+z²) aur â = a/|a|. Direction cosines l, m, n ke liye l²+m²+n² = 1 yaad rakho; direction ratios sirf components hote hain. Section formula internal ke liye (m b + n a)/(m+n), external ke liye (m b - n a)/(m-n), midpoint (a+b)/2. Dot product a\cdotb = |a||b|cosθ = a1b1+a2b2+a3b3; angle cosθ = a\cdotb/(|a||b|), projection = a\cdotb/|b|, aur perpendicular hone par dot zero. Cross product ki magnitude |a||b|sinθ, component form determinant se, parallel hone par cross zero; triangle ka area ½|a\timesb| aur parallelogram ka |a\timesb|. Exam mein pehle pucho kaam kya hai: length, angle, perpendicular, area ya perpendicular direction, phir usi row ki formula uthao.