Vectors - ACT Math

1. What is the result of adding the zero vector to any vector $\textbf{v}$?

   The vector $\textbf{v}$ itself. Zero vector is the additive identity for vector addition.

2. How are two vectors parallel in terms of their components?

   Proportional components, i.e., $a/c = b/d$. One vector is a scalar multiple of the other.

3. What is the magnitude of $\langle 3,4\rangle$?

   $5$. $\sqrt{3^2+4^2}=\sqrt{25}=5$.

4. What is the distance between points $A(x_1,y_1)$ and $B(x_2,y_2)$ using vectors?

   $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Distance equals the magnitude of the displacement vector.

5. Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.

   $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Divide by magnitude: $\|\textbf{v}\| = \sqrt{3^2 + 4^2} = 5$.

6. What is the component form of $\overrightarrow{PQ}$ for $P(-1,4)$ and $Q(2,0)$?

   $\langle 3,\;-4\rangle$. $\langle 2-(-1), 0-4\rangle = \langle 3,-4\rangle$.

7. Find the magnitude of $\mathbf{v} = \langle 3, 4 \rangle$.

   5. Using $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

8. Find the vector from point $A(1,2)$ to point $B(4,6)$.

   $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ Subtract starting point coordinates from ending point coordinates.

9. Find the unit vector along $\mathbf{v} = \langle 1, 1 \rangle$.

   $\frac{1}{\sqrt{2}} \langle 1, 1 \rangle$. Magnitude is $\sqrt{2}$, so divide by it.

10. What is the vector component form of $\mathbf{v} = 5\langle 1, 2 \rangle$?

    $\langle 5, 10 \rangle$. Distribute scalar to each component.

11. What is the direction vector of the line through $A(-2,5)$ and $B(4,1)$?

    $\langle 6,\;-4\rangle$. $\langle 4-(-2), 1-5\rangle = \langle 6,-4\rangle$.

12. Given $\textbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, find $-2\textbf{a}$.

    $\begin{bmatrix} -6 \\ -8 \end{bmatrix}$. Multiply each component by the scalar $-2$.

13. Given $\textbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, find $-2\textbf{a}$.

    $\begin{bmatrix} -6 \\ -8 \end{bmatrix}$. Multiply each component by the scalar $-2$.

14. What is the associative property of scalar multiplication with vectors?

    $(ab) \mathbf{v} = a(b \mathbf{v})$. Scalar multiplication order doesn't matter.

15. If $\mathbf{u} = \langle 3, 4 \rangle$, what is $\|\mathbf{u}\|$?

    $5$. Magnitude notation for vector $\mathbf{u}$.

16. What is the result of scalar multiplication $k \times \begin{bmatrix} a \\ b \\ \end{bmatrix}$?

    $\begin{bmatrix} ka \\ kb \\ \end{bmatrix}$. Multiply the scalar by each component.

17. What is the result of adding $\langle a,b\rangle+\langle c,d\rangle$?

    $\langle a+c,\;b+d\rangle$. Add corresponding components separately.

18. Identify the magnitude of the zero vector $\langle 0,0\rangle$.

    $0$. The zero vector has no length.

19. Determine the result of scalar multiplication: $\frac{1}{2}\langle 4, 6 \rangle$.

    $\langle 2, 3 \rangle$. Multiply each component by $\frac{1}{2}$.

20. What is the vector $\mathbf{v}$ if $\mathbf{v} = \langle 0, 0 \rangle$?

    Zero vector. Vector with no magnitude or direction.

21. What is $\| -2\langle 3,4\rangle \|$?

    $10$. $|-2| \cdot \|\langle 3,4\rangle\| = 2 \cdot 5 = 10$.

22. What is the midpoint of $A(-2,5)$ and $B(4,1)$?

    $\left(1,\;3\right)$. $\left(\frac{-2+4}{2},\frac{5+1}{2}\right)=(1,3)$.

23. What condition shows that vectors $\vec{u}$ and $\vec{v}$ are parallel (in $2$D)?

    $\vec{u}=k\vec{v}$ for some scalar $k$. One vector is a scalar multiple of the other.

24. What is the vector projection of $\textbf{a}$ onto $\textbf{b}$?

    $\frac{\textbf{a} \bullet \textbf{b}}{\text{||b||}^2} \textbf{b}$. Projects vector $\textbf{a}$ onto the direction of vector $\textbf{b}$.

25. Find the dot product of $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 1 \end{bmatrix}$.

    $11$. Calculate $(2)(4) + (3)(1) = 8 + 3 = 11$.

26. What is the component form of the vector from $A(x_1,y_1)$ to $B(x_2,y_2)$?

    $\langle x_2-x_1,\;y_2-y_1\rangle$. Subtract initial coordinates from final coordinates to get displacement.

27. For vectors $\mathbf{a}$ and $\mathbf{b}$, which property is $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$?

    Commutative property of dot product. Order doesn't matter in dot product.

28. Calculate the cross product of $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$.

    $\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$. Use the right-hand rule: $\textbf{i} \times \textbf{j} = \textbf{k}$.

29. Determine if vectors $\begin{bmatrix} 3 \\ 6 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$ are parallel.

    Yes, they are parallel. Check if $\frac{3}{2} = \frac{6}{4}$, which simplifies to $\frac{3}{2} = \frac{3}{2}$.

30. What is the zero vector in $2$D in component form?

    $\langle 0,0\rangle$. The additive identity vector in the plane.

31. Calculate the magnitude of vector \begin{bmatrix} 5 \ 12 \\ \text{endbmatrix}.

    $13$. Calculate $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

32. What does a zero vector's magnitude equal to?

    0. Zero vector has no length by definition.

33. What is the vector from $A(1,2)$ to $B(6,-1)$ in component form?

    $\langle 5,\;-3\rangle$. $\langle 6-1, -1-2\rangle = \langle 5,-3\rangle$.

34. What is the magnitude of $\mathbf{v} = \langle 6, 8 \rangle$?

    10. Using $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$.

35. What is the dot product of vectors $\textbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \end{bmatrix}$?

    $\textbf{u} \bullet \textbf{v} = ac + bd$. Multiply corresponding components and sum the results.

36. What is the magnitude of the zero vector?

    Zero. The zero vector has no length by definition.

37. Find the scalar multiple of $\mathbf{v} = \langle 2, 3 \rangle$ by $-2$.

    $\langle -4, -6 \rangle$. Multiply each component by $-2$.

38. Identify the zero vector in two dimensions.

    $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$. The additive identity vector with no magnitude or direction.

39. Determine if vectors $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$ are orthogonal.

    No, they are not orthogonal. Dot product is $1(4) + 2(5) + 3(6) = 32 \neq 0$.

40. What is the vector addition result of $\mathbf{u} = \langle 0, 1 \rangle$ and $\mathbf{v} = \langle 1, 0 \rangle$?

    $\langle 1, 1 \rangle$. Component-wise addition: $(0+1, 1+0)$.

41. What operation finds the angle between $\mathbf{a}$ and $\mathbf{b}$ if they are not zero vectors?

    Dot product. Cosine formula requires dot product calculation.

42. What is the direction of the vector $\mathbf{v} = \langle 0, 1 \rangle$?

    Positive y-axis. Unit vector pointing upward.

43. If $\mathbf{v} = \langle 2, 3 \rangle$, what is $3\mathbf{v}$?

    $\langle 6, 9 \rangle$. Multiply each component by the scalar.

44. Identify a property of dot product concerning vector orthogonality.

    Vectors are orthogonal if $\mathbf{a} \cdot \mathbf{b} = 0$. Zero dot product indicates perpendicular vectors.

45. Identify the operation used to find a vector's projection onto another.

    Dot product. Projects one vector onto another's direction.

46. What is the angle between vectors $\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}$?

    $90^\circ$. These are perpendicular unit vectors along coordinate axes.

47. Identify the scalar multiplication property for vector $\textbf{v}$ and scalar $0$.

    Result is the zero vector. Multiplying any vector by zero gives the zero vector.

48. What is the result of scalar multiplication $k\langle a,b\rangle$?

    $\langle ka,\;kb\rangle$. Multiply each component by the scalar.

49. What is the midpoint of segment $AB$ where $A(x_1,y_1)$ and $B(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right)$. Average the corresponding coordinates.

50. What is the slope of the line with direction vector $\langle 6,-4\rangle$?

    $-\frac{2}{3}$. $\frac{-4}{6}=-\frac{2}{3}$.

51. What is the result of subtracting $\langle a,b\rangle-\langle c,d\rangle$?

    $\langle a-c,\;b-d\rangle$. Subtract corresponding components separately.

52. Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$

    $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Divide by magnitude: $\|\textbf{v}\| = \sqrt{3^2 + 4^2} = 5$

53. What is the result of adding $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle 3, 4 \rangle$?

    $\langle 4, 6 \rangle$. Add corresponding components: $(1+3, 2+4)$.

54. What is the formula for the dot product of vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$?

    $a_1b_1 + a_2b_2$. Multiply corresponding components and sum them.

55. Determine if vectors $\begin{bmatrix} 3 \\ 6 \\ \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \\ \end{bmatrix}$ are parallel.

    Yes, they are parallel. Check if $\frac{3}{2} = \frac{6}{4}$, which simplifies to $\frac{3}{2} = \frac{3}{2}$.

56. Calculate the magnitude of $\mathbf{v} = \langle 1, 2, 2 \rangle$.

    $3$. Using $\sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3$.

57. What is $-3\langle 4,-2\rangle$?

    $\langle -12,\;6\rangle$. $-3\langle 4,-2\rangle = \langle -12,6\rangle$.

58. What is the zero vector in 3D space?

    $\langle 0, 0, 0 \rangle$. Origin point with no displacement.

59. What is a perpendicular direction vector to $\langle 3,-5\rangle$?

    $\langle 5,\;3\rangle$. Apply the perpendicular vector formula: $\langle -(-5),3\rangle$.

60. Which expression gives the magnitude of $k\vec{v}$ in terms of $k$ and $\|\vec{v}\|$?

    $\|k\vec{v}\|=|k|\,\|\vec{v}\|$. Scalar multiplication scales the magnitude by absolute value.

61. Find the dot product of \begin{bmatrix} 2 \ 3 \\ \text{endbmatrix} and \begin{bmatrix} 4 \ 1 \\ \text{endbmatrix}.

    $11$. Calculate $(2)(4) + (3)(1) = 8 + 3 = 11$.

62. State the distributive property for vectors and scalar multiplication.

    $c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}$. Scalar distributes over vector addition.

63. Which operation is used to determine if two vectors are perpendicular?

    Dot product. Zero result indicates perpendicular vectors.

64. What is the dot product of vectors $\textbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \end{bmatrix}$?

    $\textbf{u} \bullet \textbf{v} = ac + bd$. Multiply corresponding components and sum the results.

65. What does the dot product of parallel vectors equal to?

    Product of magnitudes. Parallel vectors have maximum dot product value.

66. State the result of the cross product of parallel vectors.

    The zero vector. Cross product of parallel vectors always equals zero.

67. What is the length of the vector $\mathbf{v} = \langle 0, 0, 0 \rangle$?

    0. Zero vector has zero magnitude.

68. State the formula to find the angle $\theta$ between vectors $\mathbf{a}$ and $\mathbf{b}$.

    $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}$. Dot product divided by product of magnitudes.

69. How do you calculate the direction angle of vector $\textbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$?

    $\theta = \tan^{-1}(\frac{b}{a})$. Use arctangent of the ratio of vertical to horizontal components.

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

    The product of magnitudes and cosine of the angle between. Relates to both magnitude and directional alignment.

71. State the condition for two vectors to be orthogonal.

    Their dot product is zero. Orthogonal vectors have perpendicular direction.

72. What is the slope of a direction vector $\langle a,b\rangle$ when $a\neq 0$?

    $\frac{b}{a}$. Rise over run: $\frac{\text{vertical component}}{\text{horizontal component}}$.

73. Express vector $\textbf{v} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}$ in terms of unit vectors $\textbf{i}$ and $\textbf{j}$.

    $4\textbf{i} - 3\textbf{j}$. Express using standard unit vectors $\textbf{i}$ and $\textbf{j}$.

74. What is $\langle 7,3\rangle-\langle 2,10\rangle$?

    $\langle 5,\;-7\rangle$. $\langle 7-2, 3-10\rangle = \langle 5,-7\rangle$.

75. What is the result of scalar multiplication $k \times \begin{bmatrix} a \\ b \end{bmatrix}$?

    $\begin{bmatrix} ka \\ kb \end{bmatrix}$. Multiply the scalar by each component.

76. What is $\|\langle -9,12\rangle\|$?

    $15$. $\sqrt{(-9)^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.

77. What is $\langle 2,-1\rangle+\langle -5,6\rangle$?

    $\langle -3,\;5\rangle$. $\langle 2+(-5), -1+6\rangle = \langle -3,5\rangle$.

78. Calculate the magnitude of vector $\begin{bmatrix} 5 \\ 12 \end{bmatrix}$.

    $13$. Calculate $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

79. Express $\textbf{v} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ in terms of unit vectors.

    $\textbf{k}$. This is the standard unit vector in the z-direction.

80. What is the unit vector in the direction of $\mathbf{v} = \langle a, b \rangle$?

    $\frac{1}{\sqrt{a^2 + b^2}} \langle a, b \rangle$. Divide vector by its magnitude to get unit length.

81. What is the magnitude formula for a vector $\mathbf{v} = \langle a, b \rangle$?

    $\sqrt{a^2 + b^2}$. Pythagorean theorem applied to vector components.

82. What is the component form of the vector $5\mathbf{i}-2\mathbf{j}$?

    $\langle 5,\;-2\rangle$. Standard unit vectors: $\mathbf{i}=\langle 1,0\rangle$, $\mathbf{j}=\langle 0,1\rangle$.

83. Identify whether $\langle 4,6\rangle$ and $\langle 2,3\rangle$ are parallel.

    Yes, because $\langle 4,6\rangle=2\langle 2,3\rangle$. One vector is a scalar multiple of the other.

84. Calculate the dot product of $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 3 \\ -4 \end{bmatrix}$.

    $-10$. Calculate $(-2)(3) + (1)(-4) = -6 - 4 = -10$.

85. What is a perpendicular direction vector to $\langle a,b\rangle$ in $2$D?

    $\langle -b,\;a\rangle$. Swap components and negate one to get perpendicular vector.

86. What is the resultant vector of $\mathbf{a} = \langle 2, 3 \rangle$ and $\mathbf{b} = \langle -2, -3 \rangle$?

    $\langle 0, 0 \rangle$. Opposite vectors sum to zero vector.

87. State the associative property for vector addition.

    $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$. Grouping doesn't affect vector addition result.

88. What is the magnitude of the vector $\vec{v}=\langle a,b\rangle$ in the plane?

    $\|\vec{v}\|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to the components.

89. Calculate the dot product of $\mathbf{a} = \langle 1, 0 \rangle$ and $\mathbf{b} = \langle 0, 1 \rangle$.

    0. $(1)(0) + (0)(1) = 0$, confirming orthogonality.

90. What is a vector's direction angle $\theta$ if $\mathbf{v} = \langle 1, 0 \rangle$?

    $0^\circ$. Vector points along positive x-axis.

91. What is the distance between $A(1,2)$ and $B(6,-1)$?

    $\sqrt{34}$. $\sqrt{(6-1)^2+(-1-2)^2}=\sqrt{25+9}=\sqrt{34}$.

92. What is the formula for the magnitude of a 3D vector $\mathbf{v} = \langle a, b, c \rangle$?

    $\sqrt{a^2 + b^2 + c^2}$. 3D extension of Pythagorean theorem.

93. How do you find the unit vector of $\textbf{v} = \begin{bmatrix} a \\ b \\ \end{bmatrix}$?

    $\frac{\textbf{v}}{||v||} = \begin{bmatrix} \frac{a}{||v||} \\ \frac{b}{||v||} \\ \end{bmatrix}$. Divide the vector by its magnitude to get length 1.

94. What is a direction vector for the line through $A(x_1,y_1)$ and $B(x_2,y_2)$?

    $\langle x_2-x_1,\;y_2-y_1\rangle$. Same as the displacement vector from $A$ to $B$.

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

    $25$. Calculate $\sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

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