Vectors and Equilibrium

Vectors and Equilibrium is the mathematical foundation of mechanics. The AAMC MCAT content outline expects fluency in adding vectors via rectangular components, computing scalar (dot) products, and computing vector (cross) products. Vector reasoning often appears as part of a hybrid mechanics problem.

Addition of Vectors (Rectangular Components)

A vector A in 2-D can be resolved into perpendicular (rectangular) components along x and y axes:

A = A_xî + A_yĵ

where î, ĵ, k̂ are the unit vectors along x, y, z. The components are:

• A_x = A·cosθ
• A_y = A·sinθ

Conversely, given the components, the magnitude and direction of A are:

• |A| = √(A_x² + A_y²)
• tanθ = A_y/A_x

Adding two vectors

To add A and B by components: split each into x and y components, add the components separately, then re-combine:

• R_x = A_x + B_x
• R_y = A_y + B_y
• |R| = √(R_x² + R_y²)
• tanφ = R_y/R_x

Scalar Product

The scalar (dot) product of two vectors A and B is a scalar quantity defined by:

A·B = |A||B|cosθ

where θ is the angle between A and B. In component form:

A·B = A_xB_x + A_yB_y + A_zB_z

Properties

• Commutative: A·B = B·A.
• Distributive: A·(B + C) = A·B + A·C.
• Self-product: A·A = |A|².
• If A ⊥ B then A·B = 0.
• If A ∥ B then A·B = |A||B|.
• Unit vectors: î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0.

Physical examples

• Work done by a constant force: W = F·d·cosθ = F·d (dot product).
• Power: P = F·v.
• Magnetic flux: Φ = B·A.

Vector Product

The vector (cross) product of A and B is a vector defined by:

A × B = |A||B|sinθ·n̂

where n̂ is a unit vector perpendicular to the plane containing A and B, with direction given by the right-hand rule: curl the fingers of the right hand from A to B; the thumb points in the direction of A × B.

Properties

• Anti-commutative: A × B = −(B × A).
• Distributive: A × (B + C) = A × B + A × C.
• Self-product: A × A = 0.
• If A ∥ B then A × B = 0.
• If A ⊥ B then |A × B| = |A||B|.
• Unit vectors (cyclic order): î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Reverse order gives a minus sign.

Physical examples

• Torque: τ = r × F.
• Angular momentum: L = r × p.
• Magnetic force on a moving charge: F = qv × B.
• Magnitude of A × B = area of the parallelogram with sides A and B.

Determinant form

For A = A_xî + A_yĵ + A_zk̂ and B = B_xî + B_yĵ + B_zk̂:

A × B = (A_yB_z − A_zB_y)î + (A_zB_x − A_xB_z)ĵ + (A_xB_y − A_yB_x)k̂

Equilibrium of Forces

A body is in equilibrium when its state of rest or uniform motion is unchanged. Two conditions must hold:

First condition (translational equilibrium)

The vector sum of all forces acting on the body is zero: ΣF = 0. Equivalent: ΣF_x = 0 and ΣF_y = 0.

Second condition (rotational equilibrium)

The vector sum of all torques about any point is zero: Στ = 0.

Both conditions together → complete equilibrium. A book resting on a table satisfies the first; a balanced see-saw satisfies both.

The MCAT-favored way to solve an equilibrium problem is to resolve every force into rectangular components and set each direction's sum to zero (ΣF_x = 0, ΣF_y = 0), then, if rotation is involved, set Στ = 0 about a convenient pivot (choosing the pivot at an unknown force eliminates it from the torque equation). Three coplanar forces in equilibrium can also be drawn head-to-tail to form a closed triangle.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter.

Quick Recap

• A = A_xî + A_yĵ; |A| = √(A_x² + A_y²); tanθ = A_y/A_x.
• Add by components: R_x = ΣA_x, R_y = ΣA_y.
• A·B = |A||B|cosθ = A_xB_x + A_yB_y; commutative.
• A × B = |A||B|sinθ·n̂; anti-commutative.
• î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ (cyclic).
• Work = F·d (dot); Torque = r × F (cross).