Vectors flash cards

1. 1.What is a scalar quantity?

   A quantity that has only magnitude (a number with a unit) and no direction. It obeys ordinary algebra. Examples: mass, time, temperature, speed, work, energy, charge.

   Hint: Magnitude only.

2. 2.What is a vector quantity?

   A quantity that has both magnitude and direction and obeys the laws of vector addition (triangle/parallelogram law). Examples: displacement, velocity, acceleration, force, momentum.

   Hint: Magnitude + direction + adds like vectors.

3. 3.Is electric current a scalar or vector? Why?

   Current is a scalar. Although it has direction, it does not obey the parallelogram law of addition (currents at a junction add algebraically per Kirchhoff), so it is a scalar.

   Hint: Direction alone doesn't make a vector.

4. 4.Distinguish speed and velocity.

   Speed is a scalar = magnitude of velocity. Velocity is a vector = displacement per unit time with direction. $|\vec{v}|$ equals instantaneous speed.

   Hint: One is |the other|.

5. 5.What is a unit vector?

   A vector of magnitude $1$ pointing in a given direction. For $\vec{A}$, the unit vector is $\hat{A}=\dfrac{\vec{A}}{|\vec{A}|}$. It carries direction only, no units of the original quantity.

   Hint: Divide a vector by its own magnitude.

6. 6.Write $\vec{A}$ in terms of its magnitude and unit vector.

   $\vec{A}=|\vec{A}|\,\hat{A}=A\,\hat{A}$, where $A=|\vec{A}|$ and $\hat{A}$ gives the direction.

   Hint: Magnitude times direction.

7. 7.What are $\hat{i},\hat{j},\hat{k}$?

   The Cartesian base unit vectors along the $x$, $y$, $z$ axes respectively. They are mutually perpendicular, each of magnitude $1$, and form a right-handed set.

   Hint: Axis directions, length 1.

8. 8.What is a null (zero) vector and one of its properties?

   $\vec{0}$ has zero magnitude and indeterminate direction. Properties: $\vec{A}+\vec{0}=\vec{A}$, $\vec{A}+(-\vec{A})=\vec{0}$, and $\lambda\vec{0}=\vec{0}$.

   Hint: Magnitude 0, direction undefined.

9. 9.What are equal vectors?

   Two vectors are equal if they have the same magnitude and same direction, regardless of their initial points (position in space). Vectors can be translated freely.

   Hint: Same length, same arrow direction.

10. 10.What is a negative vector $-\vec{A}$?

    A vector with the same magnitude as $\vec{A}$ but opposite direction. Adding it gives $\vec{A}+(-\vec{A})=\vec{0}$.

    Hint: Flip the arrow, same length.

11. 11.Define collinear (parallel/antiparallel) vectors.

    Vectors along the same line or parallel lines. Parallel: angle $0^\circ$; antiparallel: angle $180^\circ$. Any two collinear vectors satisfy $\vec{A}=\lambda\vec{B}$ for some scalar $\lambda$.

    Hint: Angle $0^\circ$ or $180^\circ$.

12. 12.What are coplanar vectors?

    Three or more vectors that lie in the same plane. Any two vectors are always coplanar. Coplanarity of $\vec{a},\vec{b},\vec{c}$ requires their scalar triple product $\vec{a}\cdot(\vec{b}\times\vec{c})=0$.

    Hint: Same plane; triple product 0.

13. 13.State the triangle law of vector addition.

    If two vectors are represented by two sides of a triangle taken in order (head-to-tail), their resultant is the third side taken in the opposite order: $\vec{R}=\vec{A}+\vec{B}$.

    Hint: Head-to-tail; close the triangle.

14. 14.State the parallelogram law of vector addition.

    If two vectors are the adjacent sides of a parallelogram drawn from a common point, the diagonal through that point gives the resultant $\vec{R}=\vec{A}+\vec{B}$.

    Hint: Diagonal from the common tail.

15. 15.State the polygon law of vector addition.

    If several vectors are represented by sides of a polygon taken in order (head-to-tail), the resultant is the closing side taken in reverse order. If the polygon closes, the resultant is $\vec{0}$.

    Hint: Chain them; closing side is the sum.

16. 16.Magnitude of resultant of $\vec{A}$ and $\vec{B}$ with angle $\theta$ between them?

    $R=\sqrt{A^2+B^2+2AB\cos\theta}$. This is the parallelogram-law formula.

    Hint: Law of cosines with a $+$ sign.

17. 17.Direction of the resultant of $\vec{A}$ and $\vec{B}$ (angle $\alpha$ from $\vec{A}$)?

    $\tan\alpha=\dfrac{B\sin\theta}{A+B\cos\theta}$, where $\theta$ is the angle between the two vectors.

    Hint: Perpendicular component over parallel component.

18. 18.Maximum and minimum resultant of two vectors of magnitudes $A$ and $B$?

    Maximum $=A+B$ (when parallel, $\theta=0^\circ$). Minimum $=|A-B|$ (when antiparallel, $\theta=180^\circ$). So $|A-B|\le R\le A+B$.

    Hint: Same direction max, opposite min.

19. 19.Resultant magnitude when $\vec{A}\perp\vec{B}$?

    $R=\sqrt{A^2+B^2}$ (since $\cos 90^\circ=0$), directed at $\tan\alpha=\dfrac{B}{A}$ from $\vec{A}$.

    Hint: Pythagoras.

20. 20.Two equal-magnitude vectors $A$ at angle $\theta$: what is $R$ and its direction?

    $R=2A\cos\dfrac{\theta}{2}$, and the resultant bisects the angle between them ($\alpha=\theta/2$).

    Hint: Half-angle formula; symmetry bisects.

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