CBSE · Class 12 · Mathematics · Chapter 10

Vector Algebra — Formula Sheet

Board Formulas

14 formulas

1.Vector in Component Form
$a = a_{1} i^+ a_{2} j^+ a_{3} k^$
Any 3-D vector has three components along the coordinate axes.
2.Magnitude of a Vector★
$∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$
Pythagoras in 3-D. Magnitude is always non-negative.
3.Unit Vector
$a^= \frac{a}{∣ a ∣}$
Same direction as a, magnitude 1. Undefined for the zero vector.
4.Direction Cosines★
$l = cos α, m = cos β, n = cos γ; l^{2} + m^{2} + n^{2} = 1$
Cosines of angles the vector makes with the x, y, z axes. Their squares sum to 1 — good sanity check.
5.Dot Product (Scalar Product)★
$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Result is a SCALAR. Zero iff a ⊥ b (or one vector is zero).
6.Angle Between Vectors★
$cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$
Rearranged dot-product formula. Always in [0, π].
7.Projection of a on b
$proj_{b} a = \frac{a \cdot b}{∣ b ∣}$
Scalar projection — length of shadow of a along direction of b. Sign can be negative.
8.Cross Product (Vector Product)★
$a \times b = i^a_{1} b_{1} j^a_{2} b_{2} k^a_{3} b_{3}$
Result is a VECTOR perpendicular to both a and b. Direction from right-hand rule.
9.Magnitude of Cross Product★
$∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$
Zero iff a and b are PARALLEL (θ = 0 or π) or one is zero.
10.Area of Parallelogram / Triangle★
$Area_{parallelogram} = ∣ a \times b ∣, Area_{triangle} = \frac{1}{2} ∣ a \times b ∣$
Cross product magnitude = area of parallelogram spanned by a and b. Triangle is half of it.
11.Unit Vector Perpendicular to Both
$n^= \frac{a \times b}{∣ a \times b ∣}$
Normalise the cross product. Direction by right-hand rule; other direction is −n̂.
12.Scalar Triple Product★
$[a b c] = a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$
Volume of parallelepiped with edges a, b, c. Zero iff the three vectors are COPLANAR.
13.Section Formula (Internal)
$r = \frac{m b + n a}{m + n}$
Point dividing AB internally in ratio m : n. Same form as coordinates section formula.
14.Non-commutativity of Cross Product
$a \times b = - b \times a$
Swapping order flips the sign. Dot product is commutative; cross product is not.