Formula Sheet

Vectors Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}

💡 Also called the modulus or length. Always non-negative. |a| = 0 only for the zero vector.

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3

💡 Result is a scalar. theta is the angle between the two vectors. Dot product is commutative: a.b = b.a.

⚠ Forgetting that the result is a scalar, not a vector.

|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta

💡 Result is a vector perpendicular to both a and b (right-hand rule). Cross product is NOT commutative: a x b = -(b x a).

⚠ Assuming a x b = b x a. The cross product is anti-commutative.

[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

💡 Result is a scalar. Cyclic permutation does not change the value: [a b c] = [b c a] = [c a b]. Swapping two vectors changes the sign.

\text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

💡 This gives a scalar (the component of a along b). For the vector projection, multiply by the unit vector of b.

⚠ Dividing by |a| instead of |b|, or forgetting the magnitude in the denominator entirely.

\text{Area} = \frac{1}{2}|\vec{a} \times \vec{b}|

💡 Here a and b are vectors representing two sides of the triangle from a common vertex.

\text{Area} = |\vec{a} \times \vec{b}|

💡 Parallelogram area is exactly twice the triangle area formed by the same two vectors.

V = |[\vec{a}\ \vec{b}\ \vec{c}]| = |\vec{a} \cdot (\vec{b} \times \vec{c})|

💡 Take the absolute value of the scalar triple product. Volume is always non-negative.

[\vec{a}\ \vec{b}\ \vec{c}] = 0

💡 Three vectors are coplanar if and only if their scalar triple product is zero. Equivalently, the 3x3 determinant of their components is zero.

#10

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

💡 Divides the line segment from A (position vector a) to B (position vector b) in the ratio m:n internally. For external division, use m:(-n).

⚠ Swapping m and n. The coefficient of b (the far point) goes with m (the near ratio).