Vector Analysis

Core Definitions

Physical quantities are distinguished by their directional requirements. Scalar quantities depend only on magnitude, while Vector quantities require both magnitude and direction.

$$\vec{V} = |V|\hat{u}$$

A vector is defined by its length multiplied by a unit vector indicating its orientation.

$\vec{V}$=Vector quantity(Variable)

$|V|$=Magnitude (Scalar length)(Variable)

$\hat{u}$=Unit vector (direction)(Dimensionless)

$|V| = 0$

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The quantity becomes a Null Vector with undefined direction.

Geometric Representation: Vectors are drawn as arrows where length represents magnitude and the tip indicates direction.

Equality: Two vectors are equal only if they share identical magnitude AND direction, regardless of starting position.

Scalar Multiplication: Multiplying a vector by a positive scalar $n$ gives a new vector $n\vec{A}$ with the same direction but $n$ times the magnitude. A negative scalar reverses the direction.

Dimensional Change: When a vector is multiplied by a scalar with its own dimensions, the result is a new physical quantity. For example, $m\vec{v}$ (mass × velocity) gives momentum — a vector with different dimensions than velocity.

A Unit Vector is a vector with magnitude exactly 1, used purely to specify direction. The unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ point along the x, y, and z axes respectively.

$$\hat{A} = \frac{\vec{A}}{|A|}$$

Divide any vector by its own magnitude to extract its direction as a unit vector.

$\hat{A}$=Unit vector in the direction of A(Dimensionless)

$\vec{A}$=Original vector(Variable)

$|A|$=Magnitude of A(Variable)

Any Direction: Unit vectors are not restricted to axes — $\hat{r}$ denotes direction along a position vector, $\hat{n}$ denotes a surface normal.

Shortcut: To find a unit vector, divide each component by the magnitude: $\hat{A} = \frac{A_x\hat{i} + A_y\hat{j} + A_z\hat{k}}{\sqrt{A_x^2 + A_y^2 + A_z^2}}$.

Vector Addition and Subtraction

The Resultant vector represents the single effect of multiple vectors combined via the Head-to-Tail Method or the Parallelogram Law.

$$\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$$

Resultant components are the algebraic sum of the individual vector components.

$R_x, R_y$=Resultant components(Variable)

$A_x, B_x$=x-components of vectors A and B(Variable)

$\hat{i}, \hat{j}$=Unit vectors along x and y axes(Dimensionless)

$\vec{A} = -\vec{B}$

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The resultant is a Null Vector.

$\vec{A} \parallel \vec{B}$ (same direction)

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Maximum resultant: $R = A + B$.

$\vec{A}$ antiparallel to $\vec{B}$

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Minimum resultant: $R = |A - B|$.

Commutative Property: The order of addition does not affect the resultant: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

Subtraction: Defined as adding a negative vector: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

Resultant Range: For two vectors of magnitudes $A$ and $B$, the resultant magnitude is bounded: $|A - B| \leq R \leq A + B$.

Quadrant Determination: The signs of $R_x$ and $R_y$ determine the quadrant. Use: Q1 ($\theta = \phi$), Q2 ($\theta = 180° - \phi$), Q3 ($\theta = 180° + \phi$), Q4 ($\theta = 360° - \phi$), where $\phi = \tan^{-1}|R_y/R_x|$.

Resolution into Components

Resolution is the process of splitting a vector into mutually perpendicular Rectangular Components using trigonometric principles.

$$A_x = A \cos \theta, \quad A_y = A \sin \theta$$

Components are calculated using the vector magnitude and the angle relative to the positive x-axis.

$A$=Vector magnitude(Variable)

$\theta$=Angle with positive x-axis(degrees/radians)

$A_x$=Horizontal projection(Variable)

$A_y$=Vertical projection(Variable)

$\theta = 0°$

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Entire vector lies along x-axis: $A_x = A$, $A_y = 0$.

$\theta = 90°$

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Entire vector lies along y-axis: $A_x = 0$, $A_y = A$.

$\theta = 45°$

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Equal components: $A_x = A_y = \frac{A}{\sqrt{2}}$.

Unit Vectors: The symbols $\hat{i}, \hat{j}, \hat{k}$ denote unit vectors along x, y, and z axes respectively.

Magnitude Recovery: Magnitude is found via Pythagorean theorem: $A = \sqrt{A_x^2 + A_y^2}$.

Direction Recovery: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$ — but check the quadrant using signs of $A_x$ and $A_y$.

3D Extension: In three dimensions, $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ with $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Position Vector

A Position Vector $\vec{r}$ describes the location of a point relative to the origin. Its rectangular components are the coordinates of the point itself.

$$\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}, \quad r = \sqrt{a^2 + b^2 + c^2}$$

The position vector points from the origin to the point P(a, b, c).

$a, b, c$=Coordinates of the point along x, y, z axes(Variable (m, km, etc.))

$r$=Distance from origin to the point(Variable)

Displacement Vector: The displacement from point A to point B is $\vec{AB} = \vec{r}_B - \vec{r}_A$.

Distance Between Points: $|AB| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2}$.

Scalar (Dot) Product

The Dot Product of two vectors yields a scalar quantity. It measures how much two vectors are aligned — geometrically, it is the product of one vector's magnitude with the Projection of the other onto it.

$$\vec{A} \cdot \vec{B} = AB\cos\theta$$

The scalar product equals the product of both magnitudes times the cosine of the angle between them.

$A, B$=Magnitudes of vectors A and B(Variable)

$\theta$=Angle between the two vectors (0° to 180°)(degrees/radians)

$\theta = 0°$ (parallel)

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$\vec{A} \cdot \vec{B} = AB$ — maximum positive value.

$\theta = 90°$ (perpendicular)

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$\vec{A} \cdot \vec{B} = 0$ — the key test for orthogonality.

$\theta = 180°$ (antiparallel)

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$\vec{A} \cdot \vec{B} = -AB$ — maximum negative value.

Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$ — order doesn't matter.

Component Form: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$ — multiply matching components and add.

Unit Vector Products: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ and $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$.

Physical Application: Work done by a force: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$.

Finding Angle: $\cos\theta = \frac{A_xB_x + A_yB_y + A_zB_z}{AB}$ — equate the two forms of the dot product.

The Projection of vector $\vec{A}$ onto vector $\vec{B}$ gives the component of $\vec{A}$ in the direction of $\vec{B}$. It is a scalar value that can be positive or negative.

$$\text{Proj}_{B} A = \frac{\vec{A} \cdot \vec{B}}{|B|} = A\cos\theta$$

The projection gives the scalar component of one vector along the direction of another.

$\vec{A} \cdot \vec{B}$=Dot product of the two vectors(Variable)

$|B|$=Magnitude of the direction vector (the one you project onto)(Variable)

$A\cos\theta$=Component of A along the direction of B(Variable)

$\theta = 0°$

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Projection equals the full magnitude $A$ — entire vector lies along $\vec{B}$.

$\theta = 90°$

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Projection is zero — no component along $\vec{B}$.

$\theta = 180°$

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Projection is $-A$ — vector points entirely opposite to $\vec{B}$.

Using Unit Vector: Projection of $\vec{A}$ on $\vec{B}$ equals $\vec{A} \cdot \hat{B}$, where $\hat{B} = \frac{\vec{B}}{|B|}$.

Sign Meaning: Positive projection means same general direction; negative means opposite.

Perpendicular Case: Projection is zero when vectors are perpendicular ($\cos 90° = 0$).

Vector (Cross) Product

The Cross Product of two vectors yields a new vector that is perpendicular to the plane containing both original vectors. Its direction follows the Right-Hand Rule.

$$\vec{A} \times \vec{B} = AB\sin\theta\,\hat{n}$$

The vector product has magnitude equal to the product of both magnitudes times the sine of the angle, directed perpendicular to the plane of A and B.

$A, B$=Magnitudes of vectors A and B(Variable)

$\theta$=Angle between A and B (0° to 180°)(degrees/radians)

$\hat{n}$=Unit vector perpendicular to the plane of A and B (via right-hand rule)(Dimensionless)

$\theta = 90°$ (perpendicular)

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$|\vec{A} \times \vec{B}| = AB$ — maximum magnitude.

$\theta = 0°$ or $180°$ (parallel/antiparallel)

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$\vec{A} \times \vec{B} = \vec{0}$ — cross product of parallel vectors is null vector.

Non-Commutative: Reversing the order flips the direction: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.

Right-Hand Rule: Curl fingers from $\vec{A}$ toward $\vec{B}$ through the smaller angle — thumb points in the direction of $\vec{A} \times \vec{B}$.

Unit Vector Products: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$. Self-products: $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}$.

Component Form: $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$. This can also be remembered as a 3×3 determinant with $\hat{i}, \hat{j}, \hat{k}$ in the first row.

Cyclic Order Shortcut: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ — follow the cycle $i \to j \to k \to i$. Reversing the cycle flips the sign.

Geometric Meaning: $|\vec{A} \times \vec{B}|$ equals the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$.

Physical Applications: Torque: $\vec{\tau} = \vec{r} \times \vec{F}$. Magnetic force: $\vec{F} = q(\vec{v} \times \vec{B})$.