Scalars & Vectors

Triangle Method (Tip-to-Tail): To add two vectors, place the tail of the second vector at the tip of the first; the resultant vector is the straight line drawn from the start of the first to the end of the second.

Parallelogram Method: When two vectors originate from the same point (tail-to-tail), they form two sides of a parallelogram; the resultant is the diagonal starting from that common origin.

Vector Subtraction: Subtracting a vector is mathematically equivalent to adding its negative. To subtract $\vec{B}$ from $\vec{A}$, one reverses the direction of $\vec{B}$ and adds it to $\vec{A}$ using standard addition rules.

Definition: Resolving a vector involves breaking it down into two perpendicular parts, typically along the x and y axes, which together have the same physical effect as the original vector.

Trigonometric Formulas: For a vector of magnitude $R$ at an angle $\theta$ to the horizontal, the components are calculated as $R_x = R \cos \theta$ and $R_y = R \sin \theta$.

Reconstruction: If the components are known, the magnitude of the resultant is found using Pythagoras' theorem: $R = \sqrt{R_x^2 + R_y^2}$, and the angle is found using $\theta = \tan^{-1}(\frac{R_y}{R_x})$.

Always Sketch: Before performing calculations, draw a rough vector diagram to visualize the direction and relative magnitudes of the components or resultants.

Check the Angle Reference: Ensure you know if the angle $\theta$ is measured from the horizontal or vertical axis, as this determines whether you use $\sin$ or $\cos$ for a specific component.

Sanity Check: The magnitude of a resultant vector must always be less than or equal to the sum of the magnitudes of its individual components, and the hypotenuse of a vector triangle must be the longest side.

Algebraic Addition Error: A common mistake is adding the magnitudes of two vectors directly (e.g., $3N + 4N = 7N$) without considering the angle between them; this is only valid if they are parallel and in the same direction.

Sign Errors in Subtraction: When subtracting vectors, students often forget to reverse the direction of the vector being subtracted before applying the addition rule.

Equilibrium Misunderstanding: In equilibrium, the resultant of all forces must be zero, which geometrically means the vectors must form a closed loop (e.g., a closed triangle).