Vector Algebra

01 / UNDERSTAND & APPLY

1. A vector combines a magnitude with a geometric direction

A scalar is specified by a signed numerical value and a unit, while an ordinary spatial vector also requires direction and obeys vector addition. Mass and energy are scalars; displacement, velocity and force are vectors. A quantity having a directional label in everyday speech does not automatically make every operation on it a vector operation. The mathematical transformation and addition rules matter.

Two free vectors are equal when their magnitudes and directions match, even if drawn with their tails at different locations. Translating a vector parallel to itself preserves the free vector. But a physical force’s point of application can matter for torque: moving the same force to a different line of action can change its effect on a rigid body. Distinguish a geometric vector from all the information in a physical loading arrangement.

The magnitude |A| is nonnegative. A nonzero vector has a unit direction  = A/|A|, which is dimensionless. The zero vector has magnitude zero and no unique direction, so it cannot be normalised by division by its magnitude. Multiplying by a negative scalar reverses direction, while multiplying by a positive scalar preserves it.

A vector equation imposes simultaneous component conditions. A + B = 0 requires B = −A, not merely equality of magnitudes. Conversely, |A| = |B| does not prove A = B. Many mechanics errors begin by replacing a vector equation with a magnitude equation before the geometry has been resolved.

02 / UNDERSTAND & APPLY

2. Resolve on perpendicular axes before combining magnitudes

In an orthonormal Cartesian basis, A = A_x î + A_y ĵ + A_z k̂. The basis vectors have unit magnitude, are mutually perpendicular and point along the positive coordinate axes. Components can be negative even though magnitude is never negative. The magnitude is √(A_x² + A_y² + A_z²) because the axes are perpendicular.

For a planar vector of magnitude A at angle θ measured counterclockwise from the positive x axis, A_x = A cosθ and A_y = A sinθ. If the angle is measured from the y axis or from a negative axis, first identify which component is adjacent and attach its sign. Memorising “x is cosine” without the angle definition is unsafe.

To recover a direction, use both component signs. The equation tanθ = A_y/A_x alone cannot distinguish opposite quadrants and fails when A_x = 0. A quadrant-aware arctangent, often called atan2(A_y,A_x), encodes that information for nonzero vectors. On paper, find a reference angle and then locate the quadrant from the sketch.

In three dimensions, direction cosines l, m, n are A_x/A, A_y/A, A_z/A for a nonzero vector, and obey l² + m² + n² = 1. They are not three independent arbitrary numbers. The angles are each measured from a positive coordinate axis, so some cosines may be negative.

03 / UNDERSTAND & APPLY

3. Head-to-tail geometry and component arithmetic are the same operation

To add vectors graphically, place the tail of the second at the head of the first without rotating it. The resultant runs from the original tail to the final head. A parallelogram construction with a common tail gives the same diagonal. Addition is commutative and associative, so any order gives the same resultant when vectors are kept unchanged.

In components, add corresponding coordinates: (A + B)_x = A_x + B_x, and similarly for y and z. Magnitudes usually do not add. For two vectors at included angle θ, |A + B|² = A² + B² + 2AB cosθ. The resultant magnitude lies between |A − B| and A + B, where A and B in those bounds denote the nonnegative magnitudes.

Subtraction is addition of the reversed vector: A − B = A + (−B). If A and B share a tail, A − B points from the head of B to the head of A. Relative position, relative velocity and changes in momentum all use this geometry. Reversing the subtraction changes direction but leaves magnitude unchanged.

A closed head-to-tail polygon means the vector sum is zero. In force equilibrium that can represent balanced forces on a particle, provided all relevant forces are included. For a rigid body, zero resultant force alone does not guarantee zero resultant torque; the polygon says nothing about the separate lines of action.

04 / UNDERSTAND & APPLY

4. The dot product measures the parallel part

The scalar product is A·B = AB cosθ, with θ the angle between nonzero vectors. In an orthonormal Cartesian basis it equals A_xB_x + A_yB_y + A_zB_z. It is commutative and distributes over addition. A·A = A² is a useful way to recover a magnitude or expand the square of a vector sum.

For nonzero A and B, a zero dot product means the vectors are perpendicular. If one vector is zero, the dot product also vanishes but no angle is defined. A positive dot product corresponds to an acute included angle; a negative value corresponds to an obtuse one. The Cauchy bound |A·B| ≤ AB provides a useful numerical check.

The scalar projection of A along a unit direction n̂ is A·n̂. The vector projection onto a nonzero B is [(A·B)/B²]B. These are different outputs: one is a signed scalar component, the other a vector along the chosen axis. The perpendicular remainder A − projection is orthogonal to B.

Work W = F·Δr for a constant force and power P = F·v are physical dot products. Units multiply along with the components: N times m gives J, and N times m s⁻¹ gives W. A zero dot product can mean force redirects motion without transferring kinetic energy, rather than meaning the force is absent.

05 / UNDERSTAND & APPLY

5. The cross product measures oriented area and perpendicular action

In three-dimensional Euclidean space, the cross product A × B has magnitude AB sinθ and direction perpendicular to both vectors by the right-hand rule. It reverses sign when the order is swapped: B × A = −A × B. For a right-handed basis, î × ĵ = k̂, ĵ × k̂ = î and k̂ × î = ĵ; reversing each pair gives the negative result.

In components, A × B = (A_yB_z − A_zB_y, A_zB_x − A_xB_z, A_xB_y − A_yB_x). The middle component is especially prone to a sign error if a determinant mnemonic is expanded carelessly. Writing the cyclic component formula and checking a simple axis case can prevent the mistake.

The magnitude is the area of the parallelogram spanned by the vectors, and half of it is the triangle area. For nonzero vectors, a zero cross product means parallel or antiparallel directions; it does not distinguish them. If either vector is zero, the cross product vanishes without defining an angle.

Torque τ = r × F uses the position vector from the chosen origin to the force application point; magnetic force qv × B uses the velocity–field order and then the sign of charge. A negative charge reverses the cross-product force direction. Replacing sinθ by cosθ confuses the perpendicular action with a parallel projection.

06 / UNDERSTAND & APPLY

6. Triple products distinguish volume from another vector

The scalar triple product A·(B × C) is the signed volume of the parallelepiped formed by the three vectors. Its absolute value gives the geometric volume. Cyclic permutations preserve it, while swapping two vectors reverses the sign. A zero value means the three vectors are linearly dependent, including the case where they lie in one plane through the common origin.

The vector triple product A × (B × C) is a vector in the plane spanned by B and C and equals B(A·C) − C(A·B). Parentheses are essential: the cross product is not associative, so (A × B) × C generally differs. The dot products on the right are scalars multiplying the indicated vectors.

For example, î × (î × ĵ) = î × k̂ = −ĵ, while (î × î) × ĵ = 0. This simple case is an immediate warning against dropping brackets. There is also no general division by a vector that would allow cancelling a cross product as if it were ordinary scalar multiplication.

If A × B = A × C with nonzero A, it follows that A × (B − C) = 0, so B − C is parallel to A. It does not force B = C. Similarly, equality of dot products with A constrains only the components along A; perpendicular components can differ. Identify the information an operation retains before attempting cancellation.

07 / UNDERSTAND & APPLY

7. Coordinates change descriptions without changing the vector

Rotating the coordinate axes changes vector components while leaving the physical vector unchanged. If planar axes rotate counterclockwise by angle α relative to the original axes, the new components are A_x′ = A_x cosα + A_y sinα and A_y′ = −A_x sinα + A_y cosα. This is a passive change of description, distinct from actively rotating the vector while holding axes fixed.

Magnitude and dot products remain invariant under a common rotation of an orthonormal basis. A·B computed before or after the coordinate change gives the same scalar. This is a useful check when choosing axes along an incline or along a convenient force: the calculation simplifies, but the physical work, angle or resultant cannot depend on the arbitrary axis choice.

Component formulas involving simple sums of squares and dot products assume an orthonormal basis. On oblique nonperpendicular axes, cross terms or a metric are required; the ordinary Cartesian formulas cannot be imported unchanged. In typical introductory mechanics, choosing perpendicular axes avoids this complication.

Use vector algebra as a modeling language: relative velocity is a subtraction, equilibrium is a zero sum, work is a projection and torque is a perpendicular moment. Knowing which operation expresses the physics is more important than applying every vector identity. Add only quantities with matching physical dimensions, even when their triples of numerical components look similar.

08 / UNDERSTAND & APPLY

8. Original practice with direction and degeneracy checks

Write components in one named basis. Keep units through products, and state when a direction or angle is undefined because a vector vanishes.

09 / UNDERSTAND & APPLY

9. Check the operation, basis, sign and units

Begin with a sketch showing positive axes and the angle reference. Resolve components, perform the required operation, and reconstruct magnitude or direction only after the algebra is complete. If the answer involves an inverse trigonometric function, restore its correct quadrant. If an angle is requested, verify that both defining vectors are nonzero.

Check a dot product against the bound AB and a cross product against AB, inspect units after multiplication, and test a simple aligned or perpendicular limit. For cross products, keep the operand order and the right-handed basis explicit. For triple products, retain brackets. These small checks catch the majority of vector mistakes before they propagate into mechanics or electromagnetism.

• Vector equality requires matching components, not just matching lengths.
• Magnitude is nonnegative; scalar components and projections can be negative.
• Dot product measures parallel projection and produces a scalar.
• Cross product measures oriented perpendicular area and depends on order.
• A coordinate change does not change the underlying physical vector.
• Zero vectors and degenerate geometries require separate angle and direction statements.