Vector Resolution & Components

Resolving a vector means writing it as a sum of components along chosen directions — usually the coordinate axes. This converts vector algebra into ordinary scalar algebra.

Concept

2D Resolution

For a vector $\vec{A}$ of magnitude $A$ making angle $\theta$ with the $x$-axis: $A_x = A\cos\theta,\qquad A_y = A\sin\theta.$

So $\vec{A} = A\cos\theta\,\hat{i} + A\sin\theta\,\hat{j}$.

3D Resolution

For $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ with magnitude $A = \sqrt{A_x^2 + A_y^2 + A_z^2}.$

The angles $\alpha, \beta, \gamma$ of $\vec{A}$ with the $x$, $y$, $z$ axes satisfy: $\cos\alpha = A_x/A,\quad \cos\beta = A_y/A,\quad \cos\gamma = A_z/A.$ $\cos\alpha, \cos\beta, \cos\gamma$ are called direction cosines $l, m, n$.

Identity: $l^2 + m^2 + n^2 = 1$.

Projection of $\vec{A}$ on Direction $\hat{n}$

$\vec{A}\cdot\hat{n}$ gives the scalar projection along $\hat{n}$ (signed).

Worked Example

Q1: A force of $20$ N is directed $30°$ above the horizontal. Find horizontal and vertical components.

Solution: $F_x = 20\cos 30° = 20(\sqrt{3}/2) \approx 17.32$ N. $F_y = 20\sin 30° = 20(0.5) = 10$ N.

Q2: Find the direction cosines of $\vec{A} = 2\hat{i} - 3\hat{j} + 6\hat{k}$.

Magnitude: $A = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$.

$l = 2/7,\quad m = -3/7,\quad n = 6/7$.

Check: $(2/7)^2 + (3/7)^2 + (6/7)^2 = (4+9+36)/49 = 1$ ✓.

Q3: A 100 N force is pulled along a slope at 37° to a wall. Find the components parallel and perpendicular to the wall.

Component perpendicular: $100\sin 37° \approx 60$ N. Component parallel: $100\cos 37° \approx 80$ N.

Why Resolve?

• Forces along independent axes can be analyzed independently (Newton's 2nd law).
• Projectile motion: independently solve $x$ and $y$ equations.
• Equilibrium: each component sum must vanish separately.

Common Confusions

• "Components must always be along Cartesian axes." — Choose any convenient orthogonal pair; tilted axes are often easier (inclined plane!).
• "$A_x = A\cos\theta$ always." — Only if $\theta$ is measured from the $x$-axis. From the $y$-axis, $A_x = A\sin\theta$.
• Signs: components can be negative if the vector points to the negative axis.

Key Takeaways

• 2D: $A_x = A\cos\theta$, $A_y = A\sin\theta$ (with $\theta$ from $x$-axis).
• 3D direction cosines satisfy $l^2 + m^2 + n^2 = 1$.
• Projection of $\vec{A}$ on $\hat{n}$ is $\vec{A}\cdot\hat{n}$.
• Choose axes wisely — often along/perpendicular to motion or applied force.