Scalars and Vectors - AP Physics C Mechanics

1. Find the magnitude of vector $\textbf{v} = \begin{bmatrix} 5 \\ 12 \end{bmatrix}$

   The magnitude is 13. Use $|\textbf{v}| = \sqrt{5^2 + 12^2} = \sqrt{169} = 13$

2. What is a vector quantity?

   A vector is a quantity with both magnitude and direction. Direction distinguishes vectors from scalars.

3. What is the zero vector?

   A vector with zero magnitude and no specific direction. It's the additive identity for vector operations.

4. What is the resultant of vectors $\textbf{A} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$?

   Resultant is $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$. Add components: $(2+1, 3+1) = (3, 4)$.

5. What is the cross product of parallel vectors?

   The cross product is zero. Parallel vectors have no perpendicular component.

6. How is the direction of a vector represented?

   By the angle it makes with a reference axis. Direction angle is measured from positive x-axis.

7. Find the magnitude of vector \textbf{v} = \begin{bmatrix} 5 \ 12 \\ \textend{bmatrix}.

   The magnitude is 13. Use $|\textbf{v}| = \sqrt{5^2 + 12^2} = \sqrt{169} = 13$.

8. Determine if acceleration is a scalar or vector quantity.

   Acceleration is a vector quantity. Acceleration has both magnitude and direction.

9. Find the magnitude of the resultant vector $\textbf{R} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.

   The magnitude is 5. Use $|\textbf{R}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

10. What is the magnitude of $\textbf{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$?

    The magnitude is 0. Zero vector has magnitude $\sqrt{0^2 + 0^2} = 0$.

11. How does a vector differ from a scalar?

    A vector has direction; a scalar does not. Direction is the key distinguishing feature.

12. What is a scalar quantity?

    A scalar is a quantity with only magnitude. It has no direction component, only size.

13. Identify the vector quantity: mass, temperature, displacement.

    Displacement is a vector quantity. Displacement has both magnitude and direction.

14. What is the geometric interpretation of the dot product?

    It measures the projection of one vector onto another. Shows how much one vector extends in another's direction.

15. Find the angle between $\textbf{A} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

    The angle is 90 degrees. Perpendicular unit vectors have dot product zero.

16. State the component form of a vector given magnitude and direction.

    $\textbf{v} = v(\text{cos} \theta \textbf{i} + \text{sin} \theta \textbf{j})$. Uses trigonometric components with magnitude $v$ and angle $\theta$.

17. What is the physical significance of the cross product?

    It gives a vector perpendicular to both original vectors. Creates a vector orthogonal to both input vectors.

18. Identify the unit vector in the direction of $\textbf{v} = \begin{bmatrix} 3 \\ 4 \\ \end{bmatrix}$.

    Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \\ \end{bmatrix}$. Divide vector by its magnitude: $\frac{\textbf{v}}{|\textbf{v}|} = \frac{\textbf{v}}{5}$.

19. Calculate the cross product of $\textbf{i} \times \textbf{k}$.

    $\textbf{i} \times \textbf{k} = -\textbf{j}$. Right-hand rule: $\textbf{i} \times \textbf{k}$ points in $-\textbf{j}$ direction.

20. Determine if force is a scalar or vector quantity.

    Force is a vector quantity. Force has both magnitude and direction.

21. Identify the scalar quantity: speed, velocity, force.

    Speed is a scalar quantity. Speed lacks direction, unlike velocity and force.

22. What is the formula for vector addition?

    $\textbf{A} + \textbf{B} = \begin{bmatrix} A_x + B_x \\ A_y + B_y \end{bmatrix}$. Add corresponding components separately.

23. What is the resultant of vectors $\textbf{A} = \begin{bmatrix} 2 \\ 3 \\ \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$?

    Resultant is $\begin{bmatrix} 3 \\ 4 \\ \end{bmatrix}$. Add components: $(2+1, 3+1) = (3, 4)$.

24. Calculate the magnitude of $\textbf{v} = \begin{bmatrix} 7 \\ 24 \end{bmatrix}$

    The magnitude is 25. Use $|\textbf{v}| = \sqrt{7^2 + 24^2} = \sqrt{625} = 25$

25. Express the vector $\textbf{v} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}$ in terms of magnitude and direction.

    Magnitude is 5; direction is $\tan^{-1}(-\frac{3}{4})$. Magnitude is $\sqrt{(-4)^2 + 3^2} = 5$; angle from positive x-axis.

26. Find the angle between \textbf{A} = \begin{bmatrix} 1 \ 0 \\ \textend{bmatrix} and \textbf{B} = \begin{bmatrix} 0 \ 1 \\ \textend{bmatrix}.

    The angle is 90 degrees. Perpendicular unit vectors have dot product zero.

27. State the formula to find the angle between two vectors $\textbf{A}$ and $\textbf{B}$.

    $\theta = \cos^{-1}\frac{\textbf{A} \bullet \textbf{B}}{|\textbf{A}||\textbf{B}|}$. Uses the dot product and magnitudes to find angle.

28. State the formula for the dot product of vectors $\textbf{A}$ and $\textbf{B}$.

    $\textbf{A} \bullet \textbf{B} = A_xB_x + A_yB_y$. Multiply corresponding components and sum them.

29. Calculate the dot product of $\textbf{A} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.

    The dot product is 11. Calculate $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.

30. What is meant by the magnitude of a vector?

    Magnitude is the length or size of the vector. Magnitude is the numerical value of vector size.

31. What is the scalar triple product $\textbf{A} \bullet (\textbf{B} \times \textbf{C})$?

    It is a scalar representing volume. Represents the volume of the parallelepiped formed.

32. What is the result of $\textbf{i} \times \textbf{i}$?

    $\textbf{i} \times \textbf{i} = 0$. Cross product of identical vectors is always zero.

33. What is the scalar product of vectors $\textbf{A}$ and $\textbf{B}$?

    The scalar product is the dot product. Scalar product and dot product are synonymous terms.

34. What is the cross product of $\textbf{i}$ and $\textbf{j}$?

    $\textbf{i} \times \textbf{j} = \textbf{k}$. Right-hand rule: $\textbf{i} \times \textbf{j}$ points in $\textbf{k}$ direction.

35. Identify the result of multiplying a vector by -1.

    It reverses the direction of the vector. Negative scalar multiplication flips the vector direction.

36. Is work a scalar or vector quantity?

    Work is a scalar quantity. Work equals force dot displacement, yielding a scalar.

37. What does it mean if two vectors are orthogonal?

    Their dot product is zero. Orthogonal means perpendicular, so no projection exists.

38. Convert the vector $\textbf{v} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ to a unit vector.

    Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Magnitude is $\sqrt{6^2 + 8^2} = 10$, so divide by 10.

39. Identify the vector form of Newton's second law.

    $\textbf{F} = m\textbf{a}$. Force vector equals mass times acceleration vector.

40. Identify the unit vector in the direction of $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.

    Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Divide vector by its magnitude: $\frac{\textbf{v}}{|\textbf{v}|} = \frac{\textbf{v}}{5}$.

41. What is the effect of multiplying a vector by a scalar?

    It changes the magnitude, not the direction. Scalar multiplication scales size but preserves direction.

Study these flashcards