Vectors — NDA Mathematics

Formulas (7)

• Position vectors and displacement vectors · Position vector → displacement
  $$\vec{AB} = \vec{b} - \vec{a} \qquad AB = |\vec{b} - \vec{a}|$$
• Addition of vectors (triangle, parallelogram, polygon laws) · Vector addition properties
  $$\vec{a} + \vec{b} = \vec{b} + \vec{a} \qquad \vec{a} - \vec{b} = \vec{a} + (-\vec{b})$$
• Scalar multiplication · Scalar multiplication
  $$|k\vec{v}| = |k|\,|\vec{v}| \qquad k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$$
• Component form: the î, ĵ, k̂ basis · Component form
  $$\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k} \qquad \vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$$
• Types of vectors (zero, unit, equal, parallel, collinear, coplanar) · Unit vector and parallelism
  $$\hat{v} = \dfrac{\vec{v}}{|\vec{v}|} \qquad \vec{a} \parallel \vec{b} \iff \vec{a} = k\vec{b} \text{ for some } k \neq 0$$
• Collinearity of three points (and vector relations in regular figures) · Collinearity test
  $$\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c} = \vec{0} \;\text{ with }\; \alpha + \beta + \gamma = 0$$
• Section Formula — Internal and External Division · Section formula (internal / external)
  $$\vec{p}_{\text{int}} = \dfrac{m\vec{b} + n\vec{a}}{m + n} \qquad \vec{p}_{\text{ext}} = \dfrac{m\vec{b} - n\vec{a}}{m - n}$$

Watch out for (14)

• An arrow drawn anywhere on the page represents the same vector
  → What is a vector? (Scalars vs vectors)
• Head minus tail — $\vec{AB} = \vec{b} - \vec{a}$, not $\vec{a} - \vec{b}$
  → Position vectors and displacement vectors
• Position vectors depend on the choice of origin; displacement vectors do NOT
  → Position vectors and displacement vectors
• Closed-polygon identity: if vectors form a closed loop, they sum to $\vec{0}$
  → Addition of vectors (triangle, parallelogram, polygon laws)
• Magnitudes don't add: $|\vec{a} + \vec{b}| \neq |\vec{a}| + |\vec{b}|$ in general
  → Addition of vectors (triangle, parallelogram, polygon laws)
• Sign of $k$ controls DIRECTION, not just signs of components
  → Scalar multiplication
• Equality of vectors = ALL components match — that's 3 equations, not 1
  → Component form: the î, ĵ, k̂ basis
• Components depend on the basis; the vector itself does not
  → Component form: the î, ĵ, k̂ basis
• Parallel VECTORS vs collinear POINTS — different conditions
  → Types of vectors (zero, unit, equal, parallel, collinear, coplanar)
• Zero vector is parallel to everything and to nothing
  → Types of vectors (zero, unit, equal, parallel, collinear, coplanar)
• Coefficient sum must be zero — don't skip the check
  → Collinearity of three points (and vector relations in regular figures)
• $(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a})=\vec{0}$ means collinear, not coplanar
  → Collinearity of three points (and vector relations in regular figures)
• External division: denominator is $m - n$, not $m + n$
  → Section Formula — Internal and External Division
• Watch the ratio order — $m : n$ means $AP : PB$, not $AP : AB$
  → Section Formula — Internal and External Division

Formulas (4)

• Magnitude of a vector and distance between two points · Magnitude and distance
  $$|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \qquad AB = |\vec{b} - \vec{a}|$$
• Direction Cosines · Direction-cosine identities
  $$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \qquad \sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$$
• Scalar projection of one vector on another · Scalar and vector projection
  $$\text{proj}_{\vec{b}}\vec{a} = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \qquad \overrightarrow{\text{proj}_{\vec{b}}\vec{a}} = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\,\vec{b}$$
• Unit vectors and direction-given construction · Unit vector and direction construction
  $$\hat{v} = \dfrac{\vec{v}}{|\vec{v}|} \qquad \vec{v} = r(\cos\alpha\,\hat{i} + \cos\beta\,\hat{j} + \cos\gamma\,\hat{k})$$

Watch out for (8)

• $\overrightarrow{AB}$ is $\vec{b} - \vec{a}$ — head minus tail
  → Magnitude of a vector and distance between two points
• Lagrange identity gives you the missing magnitude
  → Magnitude of a vector and distance between two points
• Factor-of-2 trap: $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$, not 1
  → Direction Cosines
• Direction cosines can be negative
  → Direction Cosines
• Divide by $|\vec{b}|$, not $|\vec{b}|^2$, for the scalar projection
  → Scalar projection of one vector on another
• Sign of the scalar projection encodes obtuse/acute
  → Scalar projection of one vector on another
• Check that the given angles are consistent with $\sum\cos^2 = 1$
  → Unit vectors and direction-given construction
• Equally inclined to two axes only fixes one component pair
  → Unit vectors and direction-given construction

Formulas (5)

• Dot product — components form and work done · Dot product (components form)
  $$\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \qquad W = \vec{F}\cdot\vec{d}$$
• Perpendicularity Test · Equivalent perpendicularity statements
  $$\vec{a}\perp\vec{b} \;\Longleftrightarrow\; \vec{a}\cdot\vec{b} = 0 \;\Longleftrightarrow\; |\vec{a}+\vec{b}| = |\vec{a}-\vec{b}|$$
• Angle between two vectors via the dot-product formula · Angle from dot product
  $$\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}$$
• Solving for an angle from a perpendicularity / magnitude constraint · Expansion template
  $$(\alpha\vec{a}+\beta\vec{b})\cdot(\gamma\vec{a}+\delta\vec{b}) = \alpha\gamma|\vec{a}|^2 + (\alpha\delta+\beta\gamma)\,\vec{a}\cdot\vec{b} + \beta\delta|\vec{b}|^2$$
• Unit vectors, orthogonal triples, and decomposition · Orthonormal-triple identities
  $$\vec{a}\cdot\vec{a} = 1, \quad \vec{a}\cdot\vec{b} = 0 \;\;(\text{if } \vec{a}\neq\vec{b}) \quad \text{for an orthonormal triple}$$

Watch out for (10)

• Dot product gives a scalar; cross product gives a vector
  → Dot product — components form and work done
• Work done is signed — negative work is fine
  → Dot product — components form and work done
• $|\vec{a}+\vec{b}| = |\vec{a}-\vec{b}|$ means $\vec{a}\perp\vec{b}$, not $\vec{a}=\vec{b}$
  → Perpendicularity Test
• Zero dot product needs both vectors non-zero
  → Perpendicularity Test
• Obtuse angle iff $\vec{a}\cdot\vec{b} < 0$
  → Angle between two vectors via the dot-product formula
• Direction matters when comparing two angles
  → Angle between two vectors via the dot-product formula
• Don't forget the cross terms when expanding
  → Solving for an angle from a perpendicularity / magnitude constraint
• Unit vectors mean $|\vec{a}|^2 = 1$, not $\vec{a} = 1$
  → Solving for an angle from a perpendicularity / magnitude constraint
• Three unit vectors at equal pairwise angles need not be orthonormal
  → Unit vectors, orthogonal triples, and decomposition
• $\{\vec{a}, \vec{b}, \vec{a}\times\vec{b}\}$ is orthonormal iff $\vec{a}\perp\vec{b}$ and both unit
  → Unit vectors, orthogonal triples, and decomposition

Formulas (7)

• Cross product — algebra and properties · Difference-of-squares-style identity
  $$(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = 2\,\vec{a}\times\vec{b}$$
• Cross-product magnitude, area, and the Lagrange identity · Magnitude, area, and Lagrange
  $$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta \qquad |\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2$$
• Unit vector perpendicular to two given vectors · Unit perpendicular
  $$\hat{n} = \pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}$$
• Moment of a force (torque) · Moment of a force
  $$\vec{M} = \overrightarrow{OP} \times \vec{F}$$
• Scalar triple product and coplanarity · STP as determinant + coplanarity test
  $$[\vec{a}\,\vec{b}\,\vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0 \iff \vec{a},\vec{b},\vec{c}\text{ coplanar}$$
• STP cyclic property and derived linear-combo identities · Cyclic + sum identity
  $$(\vec{a}\times\vec{b})\cdot\vec{c} + (\vec{b}\times\vec{c})\cdot\vec{a} + (\vec{c}\times\vec{a})\cdot\vec{b} = 3[\vec{a}\,\vec{b}\,\vec{c}]$$
• Vector triple product (BAC-CAB rule) · BAC-CAB rule
  $$\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\,\vec{b} - (\vec{a}\cdot\vec{b})\,\vec{c}$$

Watch out for (15)

• Cross product is NOT associative
  → Cross product — algebra and properties
• $\vec{a}\times\vec{b} = \vec{0}$ does NOT mean both vectors are zero
  → Cross product — algebra and properties
• Area of a triangle is $\tfrac{1}{2}|\vec{a}\times\vec{b}|$, NOT $|\vec{a}\times\vec{b}|$
  → Cross-product magnitude, area, and the Lagrange identity
• $\sin\theta$ is always non-negative for $\theta\in[0,\pi]$
  → Cross-product magnitude, area, and the Lagrange identity
• Both $\pm$ signs give valid answers
  → Unit vector perpendicular to two given vectors
• Scalar multiples of a unit perpendicular are not unit
  → Unit vector perpendicular to two given vectors
• Order is $\vec{r} \times \vec{F}$, not $\vec{F} \times \vec{r}$
  → Moment of a force (torque)
• Moment depends on the pivot — moment of a force about a POINT is unique, but about a LINE is also a vector
  → Moment of a force (torque)
• STP $= 0$ means coplanar — NOT \"$\vec{a}$ parallel to $\vec{b}$\"
  → Scalar triple product and coplanarity
• Determinant row/column expansion: pick the row with most zeros
  → Scalar triple product and coplanarity
• Anti-cyclic = sign flip — don't accidentally drop it
  → STP cyclic property and derived linear-combo identities
• $(\vec{a}\times\vec{b})\cdot\vec{c} = (\vec{c}\times\vec{a})\cdot\vec{b}$
  → STP cyclic property and derived linear-combo identities
• BAC-CAB only applies to vector triple products — not scalar
  → Vector triple product (BAC-CAB rule)
• Cross-then-cross is NOT cross-then-dot-with-different-grouping
  → Vector triple product (BAC-CAB rule)
• Special triples: if $\vec{a}\times\vec{b} = \vec{c}$ and $\vec{b}\times\vec{c} = \vec{a}$, the three vectors are an orthonormal pairwise-perpendicular triple
  → Vector triple product (BAC-CAB rule)

Formulas (4)

• Triangle closed-loop and centroid formula · Loop identity + centroid
  $$\vec{AB} + \vec{BC} + \vec{CA} = \vec{0} \qquad \vec{g} = \dfrac{\vec{a} + \vec{b} + \vec{c}}{3}$$
• Parallelogram properties and diagonal relations · Sides from diagonals
  $$\vec{AB} = \tfrac{1}{2}(\vec{AC} - \vec{BD}), \quad \vec{AD} = \tfrac{1}{2}(\vec{AC} + \vec{BD}), \quad \vec{OA}+\vec{OC} = \vec{OB}+\vec{OD}$$
• Angles and vertices from position vectors · Angle at vertex from position vectors
  $$\cos C = \dfrac{(\vec{a} - \vec{c})\cdot(\vec{b} - \vec{c})}{|\vec{a} - \vec{c}|\,|\vec{b} - \vec{c}|}$$
• Distance and perpendicularity identities in quadrilaterals · Parallelogram law and distance expansion
  $$|\vec{p}+\vec{q}|^2 + |\vec{p}-\vec{q}|^2 = 2(|\vec{p}|^2 + |\vec{q}|^2)$$

Watch out for (8)

• Direction matters in the loop — $\vec{BA} = -\vec{AB}$
  → Triangle closed-loop and centroid formula
• $\vec{AG}$ is NOT $\vec{g}/3$ — it is $\vec{g} - \vec{a}$
  → Triangle closed-loop and centroid formula
• Vertex order $ABCD$ matters
  → Parallelogram properties and diagonal relations
• The fourth vertex of a parallelogram: $D = A + C - B$
  → Parallelogram properties and diagonal relations
• Direction of side vectors changes the angle
  → Angles and vertices from position vectors
• Fourth-vertex problems: $D = A + C - B$, not the midpoint
  → Angles and vertices from position vectors
• Direction of comparison matters for parallelism
  → Distance and perpendicularity identities in quadrilaterals
• Expand squared distances algebraically — don't reach for coordinates first
  → Distance and perpendicularity identities in quadrilaterals