Addition of Vectors by Rectangular Components: MDCAT Physics notes

Vectors and scalars

A scalar has magnitude only; a vector has both magnitude and direction and adds by the head-to-tail (parallelogram) rule, not by simple arithmetic.

Vectors	Scalars
Displacement, velocity, acceleration	Distance, speed, time
Force, weight, momentum	Mass, work, energy, power
Electric field intensity, torque	Temperature, charge, volume

Force is a derived quantity (unit $\mathrm{kg\,m\,s^{-2}}$), not a base quantity.

Rectangular components

Any vector $\vec{A}$ making angle $\theta$ with the positive x-axis can be split into two perpendicular components:

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

so $\vec{A} = A_x\hat{i} + A_y\hat{j}$. Given the components, the magnitude and direction are

$$A = \sqrt{A_x^2 + A_y^2}, \qquad \phi = \tan^{-1}\left|\frac{A_y}{A_x}\right|$$

Adding several vectors

1. Resolve each vector into x and y components.
2. Add all x-components: $R_x = A_x + B_x + C_x + \ldots$
3. Add all y-components: $R_y = A_y + B_y + C_y + \ldots$
4. Magnitude: $R = \sqrt{R_x^2 + R_y^2}$.
5. Direction: find $\phi$ from $\tan^{-1}|R_y/R_x|$, then place it in the correct quadrant.

$R_x$	$R_y$	Quadrant	Angle $\theta$ with +x axis
+	+	I	$\phi$
−	+	II	$180^\circ - \phi$
−	−	III	$180^\circ + \phi$
+	−	IV	$360^\circ - \phi$

Worked examples

1. A force of 20 N acts at $60^\circ$ to the x-axis. $F_x = 20\cos60^\circ = 10$ N and $F_y = 20\sin60^\circ = 17.3$ N.

2. Forces of 6 N along +x and 8 N along +y act together. $R = \sqrt{36 + 64} = 10$ N, at $\tan^{-1}(8/6) \approx 53^\circ$ above the +x axis.

3. $\vec{A} = 4\hat{i} + 3\hat{j}$ and $\vec{B} = -7\hat{i} + 1\hat{j}$. $R_x = -3$, $R_y = 4$, $R = 5$. $\phi = \tan^{-1}(4/3) \approx 53^\circ$; the resultant is in quadrant II, so $\theta \approx 127^\circ$.

Key formulas

• $A_x = A\cos\theta$, $A_y = A\sin\theta$
• $R = \sqrt{R_x^2 + R_y^2}$
• $\theta = \tan^{-1}(R_y/R_x)$, adjusted for quadrant

Common MDCAT traps

• Force and displacement are vectors, not scalars or tensors.
• Components are added algebraically with signs; magnitudes are not added directly.
• $\tan^{-1}$ alone gives only the reference angle; check the signs of $R_x$ and $R_y$.
• If the angle is measured from the y-axis, the sine and cosine swap.

Quick revision

• Vectors need magnitude and direction.
• Rectangular components are perpendicular to each other.
• $A_x = A\cos\theta$, $A_y = A\sin\theta$.
• Resultant from $\sqrt{R_x^2 + R_y^2}$.
• Negative $R_x$ with positive $R_y$ puts the resultant in quadrant II.