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Formula Sheets / Maths

Chapter 28

Vector Algebra

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Vectors, addition and section formula Scalar (dot) product Vector (cross) product Triple products

Vectors, addition and section formula

Magnitude and unit vector
$a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k},$ $∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2},$ $\overset{a}{^} = \frac{a}{∣ a ∣}$
Section formula
$r = \frac{m b + n a}{m + n}$ (internal)
Collinear
$a = λ b;$ points collinear $⟺ A B = λ A C$

Scalar (dot) product

$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Projection
projection of $a$ on $b = \frac{a \cdot b}{∣ b ∣}$
$a ⊥ b ⟺ a \cdot b = 0;$ $∣ a + b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + 2 a \cdot b$

Vector (cross) product

$a \times b = ∣ a ∣∣ b ∣ sin θ \overset{n}{^} = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}$
Area
parallelogram $∣ a \times b ∣;$ triangle $\frac{1}{2} ∣ a \times b ∣$
Lagrange's identity
$∣ a \times b ∣^{2} + (a \cdot b)^{2} = ∣ a ∣^{2} ∣ b ∣^{2}$

Triple products

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 the parallelepiped; zero iff the vectors are coplanar. Tetrahedron volume is $\frac{1}{6} [a b c]$ .
Vector triple product
$a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$
$[a \times b b \times c c \times a] = [a b c]^{2}$

Hyperbola Three Dimensional Geometry