Summary

1. A scalar quantity has magnitude only — distance, speed, mass, and temperature are all scalars. A vector quantity has both magnitude and direction — displacement, velocity, and acceleration are examples, and vectors combine according to their own special rules of vector algebra (not ordinary arithmetic).

2. Multiplying a vector $\mathbf{A}$ by a real number $\lambda$ gives another vector: its magnitude becomes $\lambda$ times the magnitude of $\mathbf{A}$, and its direction stays the same as $\mathbf{A}$'s if $\lambda$ is positive, or reverses if $\lambda$ is negative.

3. Two vectors $\mathbf{A}$ and $\mathbf{B}$ can be added graphically using either the head-to-tail (triangle) method — placing the tail of $\mathbf{B}$ at the head of $\mathbf{A}$ and drawing the resultant from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$ — or the equivalent parallelogram method.

4. Vector addition is commutative, $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$, and associative, $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$ — the order in which vectors are added, or grouped, never changes the resultant.

5. A null (zero) vector $\mathbf{0}$ has zero magnitude, so its direction is undefined/immaterial. It satisfies $\mathbf{A} + \mathbf{0} = \mathbf{A}$, $\lambda\mathbf{0} = \mathbf{0}$, and $0\,\mathbf{A} = \mathbf{0}$.

6. Subtracting vector $\mathbf{B}$ from $\mathbf{A}$ means adding the negative of $\mathbf{B}$: $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$, where $-\mathbf{B}$ has the same magnitude as $\mathbf{B}$ but points in the opposite direction.

7. Any vector $\mathbf{A}$ lying in the plane of two other (non-parallel) vectors $\mathbf{a}$ and $\mathbf{b}$ can be resolved into components along them: $\mathbf{A} = \lambda\mathbf{a} + \mu\mathbf{b}$, where $\lambda$ and $\mu$ are real numbers unique to that choice of $\mathbf{a}$, $\mathbf{b}$.

8. A unit vector has magnitude exactly $1$ and points in a specific direction; the unit vector along $\mathbf{A}$ is $\hat{\mathbf{n}} = \mathbf{A}/|\mathbf{A}|$. The unit vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, $\hat{\mathbf{k}}$ point along the $x$-, $y$-, and $z$-axes respectively, in a right-handed coordinate system.

9. In component form, $\mathbf{A} = A_x\hat{\mathbf{i}} + A_y\hat{\mathbf{j}}$, where the components are $A_x = A\cos\theta$ and $A_y = A\sin\theta$ for a vector making angle $\theta$ with the $x$-axis. The magnitude and direction follow as $A = |\mathbf{A}| = \sqrt{A_x^{2} + A_y^{2}}$ and $\tan\theta = A_y/A_x$.

10. Vectors are added conveniently by the analytical (component) method: if $\mathbf{R} = \mathbf{A} + \mathbf{B}$ in the $x$-$y$ plane, then $\mathbf{R} = R_x\hat{\mathbf{i}} + R_y\hat{\mathbf{j}}$ with $R_x = A_x + B_x$ and $R_y = A_y + B_y$ — each axis adds independently.

11. The position vector of a particle in the $x$-$y$ plane is $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}}$. Its displacement on moving from $\mathbf{r}$ to $\mathbf{r}'$ is $\Delta\mathbf{r} = \mathbf{r}' - \mathbf{r} = (x' - x)\hat{\mathbf{i}} + (y' - y)\hat{\mathbf{j}} = \Delta x\,\hat{\mathbf{i}} + \Delta y\,\hat{\mathbf{j}}$.

12. Average velocity over a displacement $\Delta\mathbf{r}$ in time $\Delta t$ is $\overline{\mathbf{v}} = \Delta\mathbf{r}/\Delta t$. Instantaneous velocity is the limit of this ratio as $\Delta t \to 0$: $\mathbf{v} = d\mathbf{r}/dt = v_x\hat{\mathbf{i}} + v_y\hat{\mathbf{j}} + v_z\hat{\mathbf{k}}$, with $v_x = dx/dt$, $v_y = dy/dt$, $v_z = dz/dt$. Plotted on a graph of the particle's path, $\mathbf{v}$ always points tangent to the curve.

13. Average acceleration is $\overline{\mathbf{a}} = \Delta\mathbf{v}/\Delta t$, and instantaneous acceleration is its limiting value as $\Delta t \to 0$: $\mathbf{a} = d\mathbf{v}/dt = a_x\hat{\mathbf{i}} + a_y\hat{\mathbf{j}} + a_z\hat{\mathbf{k}}$, where $a_x = dv_x/dt$, and similarly for $a_y$, $a_z$.

14. For motion in a plane with constant acceleration $\mathbf{a}$ (magnitude $a = \sqrt{a_x^2 + a_y^2}$), starting from position $\mathbf{r}_0$ with velocity $\mathbf{v}_0$ at $t = 0$: $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2}\mathbf{a}t^2$ and $\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$. In components: $x = x_0 + v_{0x}t + \tfrac{1}{2}a_x t^2$, $y = y_0 + v_{0y}t + \tfrac{1}{2}a_y t^2$, $v_x = v_{0x} + a_x t$, $v_y = v_{0y} + a_y t$ — the plane motion is just two independent 1-D motions along perpendicular axes, superposed.

15. A projectile launched with initial speed $v_0$ at angle $\theta_0$ to the $x$-axis (initial position at the origin) follows $x = (v_0\cos\theta_0)t$, $y = (v_0\sin\theta_0)t - \tfrac{1}{2}gt^2$, with $v_x = v_0\cos\theta_0$ (constant) and $v_y = v_0\sin\theta_0 - gt$. Eliminating $t$ gives the parabolic path $y = (\tan\theta_0)x - \dfrac{gx^2}{2(v_0\cos\theta_0)^2}$. The time to maximum height is $t_m = \dfrac{v_0\sin\theta_0}{g}$, the maximum height is $h_m = \dfrac{(v_0\sin\theta_0)^2}{2g}$, and the horizontal range (where the projectile returns to $y=0$) is $R = \dfrac{v_0^2}{g}\sin 2\theta_0$.

16. Uniform circular motion is motion at constant speed $v$ along a circular path of radius $R$; the acceleration has constant magnitude $a_c = v^2/R$ (the centripetal acceleration) and always points toward the centre. The angular speed $\omega$ (rate of change of angular position) relates to $v$ by $v = \omega R$, so $a_c = \omega^2 R$ as well. If $T$ is the time period of one revolution and $\nu = 1/T$ is the frequency, then $\omega = 2\pi\nu$, $v = 2\pi\nu R$, and $a_c = 4\pi^2\nu^2 R$.

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Table 3.1 — Key quantities of this chapter, at a glance

| Physical Quantity | Symbol | Dimensions | Unit | Remark |

| :--- | :--- | :--- | :--- | :--- | …