Vectors — MHT-CET Mathematics

Formulas (6)

• Vectors and component form · Component form
  $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$
• Magnitude of a vector and distance between two points · Magnitude and distance
  $$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \qquad AB = |\vec{b} - \vec{a}|$$
• Unit vector along a given direction · Unit vector
  $$\hat{a} = \dfrac{\vec{a}}{|\vec{a}|} \qquad r\,\hat{a} = \dfrac{r}{|\vec{a}|}\,\vec{a}$$
• Magnitude of a sum from angles or perpendicularity · Magnitude of a sum (squared)
  $$|\vec{a} + \vec{b} + \vec{c}|^2 = \sum|\vec{a}|^2 + 2\sum(\vec{a}\cdot\vec{b}), \qquad \vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$$
• Magnitude of a sum with projection-equality and a perpendicular pair · Grouped expansion with a perpendicular pair
  $$|\vec{a} + \vec{b} - \vec{c}|^2 = |\vec{a}|^2 + |\vec{b} - \vec{c}|^2 + 2\,\vec{a}\cdot(\vec{b} - \vec{c})$$
• Unit vector parallel to a parallelogram's diagonal · Unit vector along a diagonal
  $$\hat{d} = \dfrac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}$$

Watch out for (11)

• A missing axis means a zero component, not a 2-D vector
  → Vectors and component form
• $\overrightarrow{AB} = \vec{b} - \vec{a}$ — head minus tail
  → Magnitude of a vector and distance between two points
• Magnitude needs every component squared
  → Magnitude of a vector and distance between two points
• Divide by the magnitude, don't subtract it
  → Unit vector along a given direction
• Add the cross terms with the factor of 2
  → Magnitude of a sum from angles or perpendicularity
• Perpendicularity conditions cancel ALL cross terms at once
  → Magnitude of a sum from angles or perpendicularity
• Take the square root at the very end
  → Magnitude of a sum from angles or perpendicularity
• "Equal projections along $\vec{a}$" means $(\vec{b}-\vec{c})\cdot\vec{a} = 0$, not $\vec{b} = \vec{c}$
  → Magnitude of a sum with projection-equality and a perpendicular pair
• Watch the sign on the vector you subtract
  → Magnitude of a sum with projection-equality and a perpendicular pair
• Diagonal $\vec{a}+\vec{b}$, not $\vec{a}-\vec{b}$
  → Unit vector parallel to a parallelogram's diagonal
• Normalise the diagonal's own magnitude, not a stray $\sqrt{77}$
  → Unit vector parallel to a parallelogram's diagonal

Formulas (5)

• Section formula — internal, external, and midpoint · Section formula (internal / external / midpoint)
  $$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m + n} \qquad \vec{r} = \frac{m\vec{b} - n\vec{a}}{m - n} \qquad \vec{r} = \frac{\vec{a} + \vec{b}}{2}$$
• Centroid and median identities · Centroid and median vector
  $$\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \qquad \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \qquad \vec{g}_{\text{tetra}} = \frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$$
• Finding the ratio, collinearity, and cevian intersection · Ratio recovery and external division
  $$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m + n} \;\Rightarrow\; \text{ratio } m:n \qquad \vec{r}_{\text{ext}} = \frac{m\vec{b} - n\vec{a}}{m - n}$$
• Incentre, orthocentre, and the angle bisector · Incentre and angle bisector
  $$\vec{I} = \frac{a\vec{a} + b\vec{b} + c\vec{c}}{a + b + c} \qquad \text{bisector of }\angle AOB \parallel \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}$$
• Triangle and parallelogram applications · Right-angle test and parallelogram diagonals
  $$\angle A = 90^\circ \iff \vec{AB}\cdot\vec{AC} = 0 \qquad \vec{PR} = \vec{PQ} + \vec{QR}, \;\; \vec{QS} = \vec{QP} + \vec{PS}$$

Watch out for (11)

• Internal adds, external subtracts
  → Section formula — internal, external, and midpoint
• Which point gets the weight $m$?
  → Section formula — internal, external, and midpoint
• Median length $\neq$ half the side it bisects
  → Centroid and median identities
• Centroid uses position vectors, not side vectors
  → Centroid and median identities
• Internal and external points use the SAME magnitude of ratio
  → Finding the ratio, collinearity, and cevian intersection
• Cevian-intersection ratio is asked along ONE cevian
  → Finding the ratio, collinearity, and cevian intersection
• Incentre weights are OPPOSITE side lengths
  → Incentre, orthocentre, and the angle bisector
• Bisector uses unit vectors — sum, not difference
  → Incentre, orthocentre, and the angle bisector
• Orthocentre $\neq$ centroid $\neq$ circumcentre
  → Incentre, orthocentre, and the angle bisector
• Equal diagonals $\to$ rectangle, perpendicular diagonals $\to$ rhombus
  → Triangle and parallelogram applications
• Right-angle test needs side-vectors FROM the vertex
  → Triangle and parallelogram applications

Formulas (10)

• Linear combination of vectors · Linear combination
  $$\vec{r} = m\vec{a} + n\vec{b} + p\vec{c} \qquad r_i = ma_i + nb_i + pc_i$$
• Collinear vectors and collinear points (one scalar) · Collinearity (one scalar)
  $$\vec{a} = k\vec{b} \qquad \dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} = \dfrac{a_3}{b_3}$$
• Coplanar vectors (two scalars) · Coplanarity (two scalars)
  $$\vec{a} = m\vec{b} + n\vec{c} \qquad [\vec{a}\ \vec{b}\ \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0$$
• Forming a combination, then normalising / measuring it · Scale a combination to a target magnitude
  $$M\,\hat{r} = \dfrac{M}{|\vec{r}|}\,\vec{r} \qquad \hat{r} = \dfrac{\vec{r}}{|\vec{r}|}$$
• Collinearity of three points (and who lies between) · Three-point collinearity
  $$\vec{QR} = k\,\vec{PQ} \quad\text{where}\quad \vec{PQ} = \vec{q} - \vec{p},\ \ \vec{QR} = \vec{r} - \vec{q}$$
• Express a vector as a combination of two others · Two-equation linear system
  $$\begin{cases} a_1 = mb_1 + nc_1 \\ a_2 = mb_2 + nc_2 \end{cases}\ \Rightarrow\ m,\ n$$
• Linear dependence vs independence (the determinant test) · Dependence ⟺ vanishing determinant
  $$x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}\ (\text{not all }0) \iff \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0$$
• Chained-collinearity systems (\"no two collinear\") · Chained collinearity
  $$\vec{a} + 2\vec{b} = t\vec{c}, \quad \vec{b} + 3\vec{c} = \lambda\vec{a} \;\Rightarrow\; t = -6,\ \lambda = -\tfrac{1}{2}$$
• A vector lying in the plane of two others · Coplanar form + a second condition
  $$\vec{v} = m\vec{b} + n\vec{c} \quad\text{with}\quad \vec{v}\cdot\vec{q} = 0 \ \text{ or }\ \dfrac{\vec{v}\cdot\vec{c}}{|\vec{c}|} = k$$
• Components of a vector against a transformed basis · Match coefficients of a basis
  $$\begin{cases} -x + y - z = 4 \\ \ \ x - y - z = 3 \\ \ \ x + y + z = 5 \end{cases}$$

Watch out for (16)

• A linear combination is built componentwise — three sums, not one
  → Linear combination of vectors
• Collinear needs ONE scalar; coplanar needs TWO — keep the count straight
  → Collinear vectors and collinear points (one scalar)
• Parallel VECTORS vs collinear POINTS
  → Collinear vectors and collinear points (one scalar)
• Coplanar / dependent ⇒ TWO scalars and a vanishing determinant
  → Coplanar vectors (two scalars)
• Build the combination BEFORE you normalise — don't normalise the parts
  → Forming a combination, then normalising / measuring it
• \"Parallel of magnitude $M$\" has TWO answers — $\pm M\hat{r}$
  → Forming a combination, then normalising / measuring it
• Use displacements that SHARE a point
  → Collinearity of three points (and who lies between)
• Solve from two equations, but ALWAYS verify with the third
  → Express a vector as a combination of two others
• Linearly dependent = coplanar = zero determinant — three names, one idea
  → Linear dependence vs independence (the determinant test)
• A second condition (like $|\vec{c}| = \sqrt{3}$) is part of the same problem
  → Linear dependence vs independence (the determinant test)
• Match coefficients only because the vectors are independent
  → Chained-collinearity systems (\"no two collinear\")
• Read the target carefully: $\vec{a}+2\vec{b}$ vs $\vec{a}+2\vec{b}+6\vec{c}$
  → Chained-collinearity systems (\"no two collinear\")
• Angle bisector uses UNIT vectors, not the raw vectors
  → A vector lying in the plane of two others
• Coplanar form first, condition second
  → A vector lying in the plane of two others
• Equating components needs an INDEPENDENT basis
  → Components of a vector against a transformed basis
• Answer the asked combination, not the raw $x, y, z$
  → Components of a vector against a transformed basis

Formulas (15)

• Dot product — the two faces of a·b · Dot product — geometric and component forms
  $$\vec a\cdot\vec b = |\vec a|\,|\vec b|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3$$
• Magnitude of a combination via the dot product · Magnitude of a linear combination
  $$|p\vec a + q\vec b|^2 = p^2|\vec a|^2 + 2pq\,(\vec a\cdot\vec b) + q^2|\vec b|^2$$
• Perpendicularity test and solving for a parameter · Perpendicularity and the perp-parameter
  $$\vec a\perp\vec b \iff \vec a\cdot\vec b = 0 \qquad (\vec a + \lambda\vec b)\perp\vec c \Rightarrow \lambda = -\frac{\vec a\cdot\vec c}{\vec b\cdot\vec c}$$
• Disguised perpendicularity — equal-diagonal and Pythagoras forms · Equivalent perpendicularity statements
  $$\vec a\perp\vec b \iff |\vec a + \vec b| = |\vec a - \vec b| \iff |\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2$$
• Angle between two vectors via cosθ · Angle from the dot product
  $$\cos\theta = \dfrac{\vec a\cdot\vec b}{|\vec a|\,|\vec b|}$$
• Angle from a unit-vector perpendicularity constraint · Expansion of a perpendicular constraint (unit vectors)
  $$(\alpha\vec a + \beta\vec b)\cdot(\gamma\vec a + \delta\vec b) = \alpha\gamma + (\alpha\delta + \beta\gamma)\cos\theta + \beta\delta = 0$$
• Scalar and vector projection · Scalar and vector projection of a on b
  $$\text{scalar} = \frac{\vec a\cdot\vec b}{|\vec b|}, \qquad \text{vector} = \frac{\vec a\cdot\vec b}{|\vec b|^2}\,\vec b$$
• Projection onto the normal of a plane · Projection on the plane normal
  $$\vec n = \vec p \times \vec q, \qquad \text{proj} = \frac{|\vec v\cdot\vec n|}{|\vec n|}$$
• Mutually orthogonal vectors — solving a small system · Mutual-orthogonality system
  $$\vec a\cdot\vec c = 0 \;\text{ and }\; \vec b\cdot\vec c = 0 \;\Rightarrow\; \text{solve for the unknown components}$$
• Unit vector along a combination, and scalar-product conditions · Unit vector of a combination
  $$\hat v = \frac{\vec v}{|\vec v|}, \qquad \frac{\vec w\cdot\vec s}{|\vec s|} = k$$
• Identities and bounds — sum of squared differences · Sum-of-squared-differences identity and bound
  $$\sum |\vec a - \vec b|^2 = 3\!\left(|\vec a|^2 + |\vec b|^2 + |\vec c|^2\right) - |\vec a + \vec b + \vec c|^2 \;\le\; 2\!\left(|\vec a|^2 + |\vec b|^2 + |\vec c|^2\right)$$
• Dot products entangled with cross-product constraints · Magnitude expansion to extract a dot product
  $$|\vec c - \vec a|^2 = |\vec c|^2 + |\vec a|^2 - 2\,\vec a\cdot\vec c \;\Rightarrow\; \vec a\cdot\vec c = \frac{|\vec c|^2 + |\vec a|^2 - |\vec c - \vec a|^2}{2}$$
• Obtuse angle for all x — a quadratic-inequality parameter · Negative-for-all-x conditions
  $$Ax^2 + Bx + C < 0 \;\forall x \iff A < 0 \;\text{ and }\; B^2 - 4AC < 0$$
• Moving point a·cos t + b·sin t — farthest from the origin · Farthest-point magnitude and direction
  $$|\overrightarrow{OP}|^2 = 1 + (\hat a\cdot\hat b)\sin 2t, \quad M = \sqrt{1 + \hat a\cdot\hat b}, \quad \hat u = \frac{\hat a + \hat b}{|\hat a + \hat b|}$$
• Reading perpendicularity geometrically — the orthocentre · Dot-zero on differences = perpendicular segments
  $$(\vec a - \vec d)\cdot(\vec b - \vec c) = 0 \iff \overrightarrow{DA}\perp\overrightarrow{BC}$$

Watch out for (33)

• Dot product is a scalar — never a vector
  → Dot product — the two faces of a·b
• Match components in the SAME direction only
  → Dot product — the two faces of a·b
• Scale FIRST, then subtract — watch the double minus
  → Magnitude of a combination via the dot product
• Magnitude is never negative
  → Magnitude of a combination via the dot product
• A missing component is ZERO, not absent
  → Perpendicularity test and solving for a parameter
• Solve for $\lambda$ cleanly: $\lambda = -(\vec a\cdot\vec c)/(\vec b\cdot\vec c)$
  → Perpendicularity test and solving for a parameter
• Order matters: $\vec a + \lambda\vec b$ vs $\vec b + \lambda\vec a$
  → Perpendicularity test and solving for a parameter
• $|\vec a + \vec b| = |\vec a - \vec b|$ is NOT $\vec a = \vec b$
  → Disguised perpendicularity — equal-diagonal and Pythagoras forms
• Pythagoras only works on the PLUS combination for a right angle
  → Disguised perpendicularity — equal-diagonal and Pythagoras forms
• Build the combinations BEFORE taking the angle
  → Angle between two vectors via cosθ
• Leave the answer as $\cos^{-1}(\cdot)$ when it is not a standard angle
  → Angle between two vectors via cosθ
• Coefficient swap flips the angle: check WHICH combination
  → Angle from a unit-vector perpendicularity constraint
• Unit vectors mean $|\vec a|^2 = 1$, not $0$
  → Angle from a unit-vector perpendicularity constraint
• Don't drop a cross term — there are TWO middle products
  → Angle from a unit-vector perpendicularity constraint
• Divide by $|\vec b|$ — projection is NOT just the dot product
  → Scalar and vector projection
• Scalar projection can be negative; magnitude of projection is its absolute value
  → Scalar and vector projection
• Vector projection divides by $|\vec b|^2$, then multiplies by $\vec b$
  → Scalar and vector projection
• 'Perpendicular to the plane' means CROSS product, not dot
  → Projection onto the normal of a plane
• Project onto the NORMAL, then divide by $|\vec n|$
  → Projection onto the normal of a plane
• Two perpendicularity conditions → two equations, not one
  → Mutually orthogonal vectors — solving a small system
• Watch sign and ordering in the answer pair
  → Mutually orthogonal vectors — solving a small system
• Divide the WHOLE vector by its magnitude
  → Unit vector along a combination, and scalar-product conditions
• Compute the combination's components BEFORE the magnitude
  → Unit vector along a combination, and scalar-product conditions
• 'Does not exceed' = maximum, not the typical value
  → Identities and bounds — sum of squared differences
• Each magnitude-squared appears TWICE in the expanded sum
  → Identities and bounds — sum of squared differences
• Extract $|\vec c|$ FIRST from the cross/angle condition
  → Dot products entangled with cross-product constraints
• $\vec b\times\vec c = \vec b\times\vec a$ is NOT $\vec c = \vec a$
  → Dot products entangled with cross-product constraints
• Obtuse-for-ALL-x needs BOTH conditions
  → Obtuse angle for all x — a quadratic-inequality parameter
• Obtuse is strict: exclude the perpendicular boundary
  → Obtuse angle for all x — a quadratic-inequality parameter
• Maximise $|\overrightarrow{OP}|^2$, then the direction is $\hat a + \hat b$ (plus, not minus)
  → Moving point a·cos t + b·sin t — farthest from the origin
• The cross term is $(\hat a\cdot\hat b)\sin 2t$, coefficient 1 not 2
  → Moving point a·cos t + b·sin t — farthest from the origin
• Altitudes → orthocentre, not circumcentre
  → Reading perpendicularity geometrically — the orthocentre
• Translate each dot-zero into the correct perpendicular pair
  → Reading perpendicularity geometrically — the orthocentre

Formulas (12)

• The cross product — definition and determinant form · Cross product as a determinant
  $$\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$$
• Magnitude of the cross product, angle, and the Lagrange identity · Magnitude and Lagrange
  $$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta \qquad |\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$$
• Area of a triangle from two side vectors · Triangle area
  $$\text{Area} = \tfrac{1}{2}\,|\overrightarrow{AB} \times \overrightarrow{AC}|$$
• Area of a parallelogram — from sides, diagonals, or a side and a diagonal · Parallelogram areas
  $$\text{Area} = |\vec{a}\times\vec{b}| \qquad \text{Area}_{\text{diagonals}} = \tfrac{1}{2}|\vec{d_1}\times\vec{d_2}|$$
• Bilinear expansion and area-scaling identities · Bilinear cross-expansion
  $$(p\vec{a} + q\vec{b}) \times (r\vec{a} + s\vec{b}) = (ps - qr)\,(\vec{a}\times\vec{b})$$
• Unit (and given-magnitude) vector perpendicular to two vectors · Unit / scaled perpendicular
  $$\hat{n} = \pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|} \qquad \vec{v}_{|m|} = \pm\, m\,\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}$$
• Solving a vector equation: a cross condition plus a scalar condition · Cross plus scalar condition
  $$\vec{a}\times\vec{r} = \vec{b}, \;\; \vec{a}\cdot\vec{r} = k \;\;\Longrightarrow\;\; \vec{r}\text{ is uniquely determined}$$
• Finding unknown components from a given cross product · Component matching
  $$\vec{b}\times\vec{c} = (\text{given vector}) \;\Longrightarrow\; \text{equate each of }\hat{i},\hat{j},\hat{k}\text{ components}$$
• Parallelism, collinearity, and a vector along a×b · Parallel via zero cross product
  $$\vec{a}\times\vec{b} = k(\vec{a}\times\vec{c}) \;\Longrightarrow\; \vec{a}\times(\vec{b} - k\vec{c}) = \vec{0} \;\Longrightarrow\; \vec{b} - k\vec{c} = \lambda\vec{a}$$
• Vector triple product — the BAC-CAB rule · BAC-CAB rule
  $$\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\,\vec{b} - (\vec{a}\cdot\vec{b})\,\vec{c}$$
• Magnitude of a vector triple product with a given angle · Triple-product magnitude
  $$|(\vec{a}\times\vec{b})\times\vec{c}| = |\vec{a}\times\vec{b}|\,|\vec{c}|\sin\phi$$
• Angle and cross-magnitude from a vector constraint · Perpendicularity of a cross product
  $$\vec{a}\cdot(\vec{a}\times\vec{b}) = 0 \qquad \vec{b}\cdot(\vec{a}\times\vec{b}) = 0$$

Watch out for (28)

• The cross product is a VECTOR, not a scalar
  → The cross product — definition and determinant form
• Watch the SIGN on the $\hat{j}$-component
  → The cross product — definition and determinant form
• $\vec{a}\times\vec{b} = \vec{0}$ does NOT mean both vectors are zero
  → The cross product — definition and determinant form
• $\sin\theta$ is the same for $\theta$ and $180^\circ - \theta$
  → Magnitude of the cross product, angle, and the Lagrange identity
• Use Lagrange to skip finding the angle
  → Magnitude of the cross product, angle, and the Lagrange identity
• The HALF is on the triangle, not the parallelogram
  → Area of a triangle from two side vectors
• When area is GIVEN, expect TWO values of the unknown
  → Area of a triangle from two side vectors
• Cross edges from the SAME vertex
  → Area of a triangle from two side vectors
• SIDES use no $\tfrac{1}{2}$; DIAGONALS do
  → Area of a parallelogram — from sides, diagonals, or a side and a diagonal
• A diagonal is the SUM of the two sides, not one of them
  → Area of a parallelogram — from sides, diagonals, or a side and a diagonal
• Keep the cross-terms in order
  → Bilinear expansion and area-scaling identities
• Area takes the ABSOLUTE value of the coefficient
  → Bilinear expansion and area-scaling identities
• Both $\pm$ signs are valid answers
  → Unit (and given-magnitude) vector perpendicular to two vectors
• For perpendicular to $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, use the shortcut
  → Unit (and given-magnitude) vector perpendicular to two vectors
• Confirm the magnitude is actually $1$
  → Unit (and given-magnitude) vector perpendicular to two vectors
• One cross equation is NOT enough on its own
  → Solving a vector equation: a cross condition plus a scalar condition
• $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ does NOT mean $\vec{b} = \vec{c}$
  → Solving a vector equation: a cross condition plus a scalar condition
• Pick the component that isolates the unknown
  → Finding unknown components from a given cross product
• Use the right datum for the right unknown
  → Finding unknown components from a given cross product
• Track the sign of the dot product
  → Parallelism, collinearity, and a vector along a×b
• Normal parallel means PLANES parallel, not perpendicular
  → Parallelism, collinearity, and a vector along a×b
• Grouping matters — the two triple products differ
  → Vector triple product — the BAC-CAB rule
• BAC-CAB is for VECTOR triple products only
  → Vector triple product — the BAC-CAB rule
• When comparing a×(a×c) problems, isolate $\vec{a}\cdot\vec{c}$
  → Vector triple product — the BAC-CAB rule
• Compute $|\vec{a}\times\vec{b}|$ FIRST, then treat it as one vector
  → Magnitude of a vector triple product with a given angle
• Recover $|\vec{c}|$ from the side conditions before using $\sin\phi$
  → Magnitude of a vector triple product with a given angle
• A cross product contributes ZERO to a dot with its own factor
  → Angle and cross-magnitude from a vector constraint
• Square the constraint to get dot products
  → Angle and cross-magnitude from a vector constraint

Formulas (11)

• The scalar triple product — dot-cross and determinant form · Scalar triple product as a determinant
  $$[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$
• Cyclic and sign properties of the scalar triple product · Cyclic and swap rules
  $$[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}] = -[\vec{b}\ \vec{a}\ \vec{c}]$$
• Computing the value of a scalar triple product · STP when one vector is perpendicular to the other two
  $$[\vec{a}\ \vec{b}\ \vec{c}] = |\vec{a}|\,|\vec{b}\times\vec{c}| = |\vec{a}||\vec{b}||\vec{c}|\sin\theta$$
• Coplanarity of three vectors (and solving for a parameter) · Coplanarity criterion
  $$\vec{a},\vec{b},\vec{c}\text{ coplanar} \iff \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0$$
• Scalar triple product of linear combinations · Linear-combination identities
  $$[\vec{a}+\vec{b}\ \ \vec{b}+\vec{c}\ \ \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}] \qquad [m\vec{a}+\vec{b}\ \ m\vec{b}+\vec{c}\ \ m\vec{c}+\vec{a}] = (m^3+1)[\vec{a}\ \vec{b}\ \vec{c}]$$
• Volume of a parallelepiped (and min/max problems) · Parallelepiped volume
  $$V = |[\vec{a}\ \vec{b}\ \vec{c}]| = \left|\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}\right|$$
• Volume of a tetrahedron · Tetrahedron volume
  $$V = \tfrac{1}{6}\,\bigl|[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}]\bigr|$$
• Reciprocal-basis identities and the STP-squared rule · Reciprocal pairings and STP-squared
  $$\vec{a}\cdot\vec{p} = \vec{b}\cdot\vec{q} = \vec{c}\cdot\vec{r} = 1, \quad \vec{a}\cdot\vec{q} = \vec{b}\cdot\vec{p} = \dots = 0 \qquad [\vec{a}\times\vec{b}\ \ \vec{b}\times\vec{c}\ \ \vec{c}\times\vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2$$
• Vector triple product (BAC-CAB rule) · BAC-CAB rule
  $$\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\,\vec{b} - (\vec{a}\cdot\vec{b})\,\vec{c}$$
• Vector orthogonal to one vector and coplanar with two others · Orthogonal-and-coplanar vector
  $$\vec{c}\times(\vec{a}\times\vec{b}) = (\vec{c}\cdot\vec{b})\,\vec{a} - (\vec{c}\cdot\vec{a})\,\vec{b}$$
• Solving a vector equation: a cross condition plus a magnitude/dot condition · Cross condition reduces to a parallel offset
  $$\vec{r}\times\vec{b} = \vec{c}\times\vec{b} \;\Longrightarrow\; \vec{r} = \vec{c} + t\,\vec{b}, \quad\text{then use the scalar condition for } t$$

Watch out for (24)

• The scalar triple product is a SCALAR
  → The scalar triple product — dot-cross and determinant form
• Dot and cross can swap, but keep the order of the three vectors
  → The scalar triple product — dot-cross and determinant form
• Cyclic keeps the value; ANY single swap negates it
  → Cyclic and sign properties of the scalar triple product
• A repeated vector kills the product — spot it early
  → Cyclic and sign properties of the scalar triple product
• Perpendicular to BOTH means parallel to the cross product
  → Computing the value of a scalar triple product
• "Depends on x and y" — expand the determinant first
  → Computing the value of a scalar triple product
• Coplanar ⇒ STP = 0, NOT "two of them are parallel"
  → Coplanarity of three vectors (and solving for a parameter)
• A squared parameter can give TWO coplanarity values
  → Coplanarity of three vectors (and solving for a parameter)
• $[m\vec{a}+\vec{b}\ \dots] = (m^3+1)[\vec{a}\ \vec{b}\ \vec{c}]$, not $m^3$
  → Scalar triple product of linear combinations
• Track every sign through the swaps
  → Scalar triple product of linear combinations
• Volume is the MODULUS — never a negative number
  → Volume of a parallelepiped (and min/max problems)
• Min vs max: check the second derivative
  → Volume of a parallelepiped (and min/max problems)
• The one-sixth is on the tetrahedron, not the parallelepiped
  → Volume of a tetrahedron
• Build all three edges from the SAME vertex
  → Volume of a tetrahedron
• Matched pairs are 1, mismatched pairs are 0
  → Reciprocal-basis identities and the STP-squared rule
• Cross-of-pairs box is the SQUARE, not the cube
  → Reciprocal-basis identities and the STP-squared rule
• Inner pair sets the plane: $\vec{a}\times(\vec{b}\times\vec{c})$ is in the $\vec{b}$-$\vec{c}$ plane
  → Vector triple product (BAC-CAB rule)
• Match coefficients only when the basis vectors are independent
  → Vector triple product (BAC-CAB rule)
• Two crosses ⇒ BAC-CAB; one cross + one dot ⇒ scalar triple product
  → Vector triple product (BAC-CAB rule)
• Coplanar means a COMBINATION, not just "in the same plane"
  → Vector orthogonal to one vector and coplanar with two others
• Both signs of the UNIT answer are valid
  → Vector orthogonal to one vector and coplanar with two others
• Confirm orthogonality AND coplanarity at the end
  → Vector orthogonal to one vector and coplanar with two others
• You cannot cancel the cross product
  → Solving a vector equation: a cross condition plus a magnitude/dot condition
• The cross condition alone leaves one free scalar
  → Solving a vector equation: a cross condition plus a magnitude/dot condition