Formula Sheet: Vectors and Scalars

Vector and Scalar Fundamentals

Definitions

• Scalar: A quantity with magnitude only (e.g., mass, temperature, speed, energy, time)
• Vector: A quantity with both magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum)
• Magnitude of a vector: The length or size of the vector, denoted as \(|\vec{A}|\) or simply \(A\)
• Unit vector: A vector with magnitude of 1, used to indicate direction; denoted as \(\hat{A} = \frac{\vec{A}}{|\vec{A}|}\)

Vector Representation

Component Form

Two-dimensional vector:

\[\vec{A} = A_x\hat{i} + A_y\hat{j}\]

• \(A_x\) = x-component of vector A
• \(A_y\) = y-component of vector A
• \(\hat{i}\) = unit vector in x-direction
• \(\hat{j}\) = unit vector in y-direction

Three-dimensional vector:

\[\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\]

• \(A_z\) = z-component of vector A
• \(\hat{k}\) = unit vector in z-direction

Magnitude Calculation

Two-dimensional magnitude:

\[|\vec{A}| = \sqrt{A_x^2 + A_y^2}\]

Three-dimensional magnitude:

\[|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}\]

Direction and Components

Components from magnitude and angle:

\[A_x = A\cos\theta\] \[A_y = A\sin\theta\]

• \(\theta\) = angle measured from positive x-axis (counterclockwise)
• \(A\) = magnitude of vector

Angle from components:

\[\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)\]

• Note: Use quadrant analysis to determine correct angle (0° to 360°)

Vector Operations

Vector Addition

Component method (most useful for MCAT):

\[\vec{R} = \vec{A} + \vec{B}\] \[R_x = A_x + B_x\] \[R_y = A_y + B_y\] \[R_z = A_z + B_z\]

• \(\vec{R}\) = resultant vector
• Add corresponding components separately

Vector Subtraction

\[\vec{R} = \vec{A} - \vec{B}\] \[R_x = A_x - B_x\] \[R_y = A_y - B_y\] \[R_z = A_z - B_z\]

• Equivalent to adding the negative: \(\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\)

Scalar Multiplication

\[c\vec{A} = cA_x\hat{i} + cA_y\hat{j} + cA_z\hat{k}\]

• \(c\) = scalar constant
• Multiplies magnitude by \(|c|\)
• Direction reversed if \(c <>
• Magnitude of result: \(|c\vec{A}| = |c| \cdot |\vec{A}|\)

Dot Product (Scalar Product)

Definition:

\[\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta\]

• \(\theta\) = angle between vectors A and B (0° ≤ θ ≤ 180°)
• Result is a scalar
• Units: product of the units of A and B

Component form:

\[\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\]

Special cases:

• If vectors are parallel (\(\theta = 0°\)): \(\vec{A} \cdot \vec{B} = AB\)
• If vectors are perpendicular (\(\theta = 90°\)): \(\vec{A} \cdot \vec{B} = 0\)
• If vectors are antiparallel (\(\theta = 180°\)): \(\vec{A} \cdot \vec{B} = -AB\)

Finding angle between vectors:

\[\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}\]

Properties:

• Commutative: \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\)
• Distributive: \(\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)
• Dot product with itself: \(\vec{A} \cdot \vec{A} = A^2\)

Cross Product (Vector Product)

Magnitude:

\[|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta\]

• \(\theta\) = angle between vectors A and B (0° ≤ θ ≤ 180°)
• Result is a vector
• Direction: perpendicular to both A and B (right-hand rule)

Component form:

\[\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}\]

Determinant form:

\[\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}\]

Special cases:

• If vectors are parallel (\(\theta = 0°\) or 180°): \(\vec{A} \times \vec{B} = 0\)
• If vectors are perpendicular (\(\theta = 90°\)): \(|\vec{A} \times \vec{B}| = AB\)

Properties:

• Anti-commutative: \(\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}\)
• Distributive: \(\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}\)
• Cross product with itself: \(\vec{A} \times \vec{A} = 0\)

Right-hand rule for direction:

• Point fingers of right hand along first vector (A)
• Curl fingers toward second vector (B)
• Thumb points in direction of \(\vec{A} \times \vec{B}\)

Applications in Physics

Displacement and Position

Displacement vector:

\[\vec{d} = \vec{r}_f - \vec{r}_i\]

• \(\vec{r}_i\) = initial position vector
• \(\vec{r}_f\) = final position vector
• Displacement is a vector quantity
• Distance is the scalar magnitude: \(d = |\vec{d}|\)

Velocity

Average velocity:

\[\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t} = \frac{\vec{r}_f - \vec{r}_i}{t_f - t_i}\]

• Vector quantity with direction of displacement
• Units: m/s

Speed vs. velocity:

• Speed: scalar, magnitude of velocity
• Velocity: vector, includes direction

Velocity components:

\[\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}\] \[|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\]

Acceleration

Average acceleration:

\[\vec{a}_{avg} = \frac{\Delta\vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i}\]

• Vector quantity
• Units: m/s²

Acceleration components:

\[\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}\]

Force

Net force (vector sum):

\[\vec{F}_{net} = \sum\vec{F}_i = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + ...\]

• Add forces as vectors (component method)
• Units: N (Newtons)

Newton's Second Law (vector form):

\[\vec{F}_{net} = m\vec{a}\]

• \(m\) = mass (scalar, kg)
• \(\vec{a}\) = acceleration vector (m/s²)

Momentum

Linear momentum:

\[\vec{p} = m\vec{v}\]

• Vector quantity in same direction as velocity
• Units: kg·m/s

Conservation of momentum (vector form):

\[\sum\vec{p}_i = \sum\vec{p}_f\] \[\vec{p}_{1i} + \vec{p}_{2i} = \vec{p}_{1f} + \vec{p}_{2f}\]

• Must be applied component-wise in different directions

Torque

Torque as cross product:

\[\vec{\tau} = \vec{r} \times \vec{F}\]

• \(\vec{r}\) = position vector from axis of rotation to point of force application
• \(\vec{F}\) = force vector
• Direction: perpendicular to plane containing r and F (right-hand rule)
• Units: N·m

Magnitude of torque:

\[|\vec{\tau}| = rF\sin\theta\] \[\tau = rF_{\perp} = r_{\perp}F\]

• \(\theta\) = angle between r and F
• \(F_{\perp}\) = component of force perpendicular to r
• \(r_{\perp}\) = perpendicular distance from axis to line of action of force (lever arm)

Angular Momentum

Angular momentum:

\[\vec{L} = \vec{r} \times \vec{p}\]

• \(\vec{r}\) = position vector from axis
• \(\vec{p}\) = linear momentum vector
• Units: kg·m²/s

Magnitude:

\[L = rp\sin\theta = mvr\sin\theta\]

• \(\theta\) = angle between r and p (or r and v)

Work (Using Dot Product)

Work by constant force:

\[W = \vec{F} \cdot \vec{d} = Fd\cos\theta\]

• \(\vec{F}\) = force vector
• \(\vec{d}\) = displacement vector
• \(\theta\) = angle between force and displacement
• Work is a scalar quantity
• Units: J (Joules) = N·m

Component form:

\[W = F_xd_x + F_yd_y + F_zd_z\]

Special cases:

• Force parallel to displacement (\(\theta = 0°\)): \(W = Fd\) (maximum positive work)
• Force perpendicular to displacement (\(\theta = 90°\)): \(W = 0\) (no work done)
• Force opposite to displacement (\(\theta = 180°\)): \(W = -Fd\) (negative work)

Projectile Motion

Decomposition into components:

• Horizontal component (x): constant velocity, no acceleration
• Vertical component (y): constant acceleration due to gravity

Initial velocity components:

\[v_{0x} = v_0\cos\theta\] \[v_{0y} = v_0\sin\theta\]

• \(v_0\) = initial speed
• \(\theta\) = launch angle above horizontal

Position vector:

\[\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}\] \[x(t) = v_{0x}t = v_0\cos\theta \cdot t\] \[y(t) = v_{0y}t - \frac{1}{2}gt^2 = v_0\sin\theta \cdot t - \frac{1}{2}gt^2\]

Velocity vector:

\[\vec{v}(t) = v_x(t)\hat{i} + v_y(t)\hat{j}\] \[v_x(t) = v_{0x} = v_0\cos\theta\] \[v_y(t) = v_{0y} - gt = v_0\sin\theta - gt\]

Inclined Planes

Coordinate system rotation:

• Choose x-axis parallel to incline, y-axis perpendicular to incline
• Decompose weight vector into components

Weight components:

\[W_{\parallel} = mg\sin\theta\] \[W_{\perp} = mg\cos\theta\]

• \(W_{\parallel}\) = component parallel to incline (down the slope)
• \(W_{\perp}\) = component perpendicular to incline (into the surface)
• \(\theta\) = angle of incline from horizontal
• \(m\) = mass (kg)
• \(g\) = acceleration due to gravity (9.8 m/s²)

Vector Resolution Summary

Key Principles

• Independence of components: Vector components along perpendicular axes are independent
• Resultant magnitude: Always use Pythagorean theorem for perpendicular components
• Direction: Use inverse tangent with quadrant consideration
• Vector addition: Add components separately, then recombine
• Dot product: Yields scalar; measures parallel components
• Cross product: Yields vector; perpendicular to both original vectors

Common MCAT Vector Applications

• Kinematics: Position, velocity, acceleration vectors
• Dynamics: Force vectors, net force calculations
• Energy: Work as dot product of force and displacement
• Momentum: Conservation in multiple dimensions
• Rotation: Torque and angular momentum as cross products
• Projectiles: Decomposition into horizontal and vertical components
• Inclines: Force resolution parallel and perpendicular to surface

Important Reminders

• Vectors can only be added to or subtracted from other vectors
• Scalars can only be added to or subtracted from other scalars
• A vector can be multiplied by a scalar to change its magnitude
• Two vectors can be multiplied using dot product (scalar result) or cross product (vector result)
• Always maintain consistent units throughout calculations
• Pay attention to angle measurement (from x-axis, from vertical, etc.)
• Use component method for most MCAT problems involving vector addition/subtraction