ACT Math Flashcards: Vectors

What this deck covers

This deck focuses on Vectors, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

Flashcard 3: What is a vector's direction angle $\theta$ if $\mathbf{v} = \langle 1, 0 \rangle$?

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

Flashcard 4: What is the associative property of scalar multiplication with vectors?

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

Flashcard 5: What is the vector addition result of $\mathbf{u} = \langle 0, 1 \rangle$ and $\mathbf{v} = \langle 1, 0 \rangle$?

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

Flashcard 6: What is the geometric interpretation of the dot product?

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

Flashcard 7: What is the magnitude of $\langle 3,4\rangle$?

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

Flashcard 8: What is the distance between points $A(x_1,y_1)$ and $B(x_2,y_2)$ using vectors?

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

Flashcard 9: What is the slope of the line with direction vector $\langle 6,-4\rangle$?

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

Flashcard 10: Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.

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

Flashcard 11: What is the component form of $\overrightarrow{PQ}$ for $P(-1,4)$ and $Q(2,0)$?

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

Flashcard 12: Find the magnitude of $\mathbf{v} = \langle 3, 4 \rangle$.

Answer:

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

Flashcard 13: What is the vector projection of $\textbf{a}$ onto $\textbf{b}$?

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

Flashcard 14: What is the direction vector of the line through $A(-2,5)$ and $B(4,1)$?

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

Flashcard 15: Find the vector from point $A(1,2)$ to point $B(4,6)$.

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

Flashcard 16: What is $\langle 7,3\rangle-\langle 2,10\rangle$?

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

Flashcard 17: Find the unit vector along $\mathbf{v} = \langle 1, 1 \rangle$.

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

Flashcard 18: Express vector $\textbf{v} = \begin{bmatrix} 4 \ -3 \end{bmatrix}$ in terms of unit vectors $\textbf{i}$ and $\textbf{j}$.

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

Flashcard 19: What is the zero vector in 3D space?

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

Flashcard 20: What is the vector component form of $\mathbf{v} = 5\langle 1, 2 \rangle$?

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

Flashcard 21: What is the direction vector of the line through $A(-2,5)$ and $B(4,1)$?

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

Flashcard 22: Given $\textbf{a} = \begin{bmatrix} 3 \ 4 \end{bmatrix}$, find $-2\textbf{a}$.

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

Flashcard 23: What is the formula for the magnitude of a 3D vector $\mathbf{v} = \langle a, b, c \rangle$?

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

Flashcard 24: Given $\textbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, find $-2\textbf{a}$.

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

Flashcard 25: What is the associative property of scalar multiplication with vectors?

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

Flashcard 26: If $\mathbf{u} = \langle 3, 4 \rangle$, what is $\|\mathbf{u}\|$?

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

Flashcard 27: What is the component form of the vector $5\mathbf{i}-2\mathbf{j}$?

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

Flashcard 28: What is the result of scalar multiplication $k \times \begin{bmatrix} a \ b \\ \end{bmatrix}$?

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

Flashcard 29: What is the magnitude of the zero vector?

Answer: Zero. The zero vector has no length by definition.

Flashcard 30: What is the result of adding $\langle a,b\rangle+\langle c,d\rangle$?

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

Flashcard 31: Express $\textbf{v} = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}$ in terms of unit vectors.

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

Flashcard 32: Identify the magnitude of the zero vector $\langle 0,0\rangle$.

Answer: $0$. The zero vector has no length.

Flashcard 33: Determine the result of scalar multiplication: $\frac{1}{2}\langle 4, 6 \rangle$.

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

Flashcard 34: What does the dot product of parallel vectors equal to?

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

Flashcard 35: What is the vector $\mathbf{v}$ if $\mathbf{v} = \langle 0, 0 \rangle$?

Answer: Zero vector. Vector with no magnitude or direction.

Flashcard 36: What is the vector $\mathbf{v}$ if $\mathbf{v} = \langle 0, 0 \rangle$?

Answer: Zero vector. Vector with no magnitude or direction.

Flashcard 37: What is the resultant vector of $\mathbf{a} = \langle 2, 3 \rangle$ and $\mathbf{b} = \langle -2, -3 \rangle$?

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

Flashcard 38: What is $\| -2\langle 3,4\rangle \|$?

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

Flashcard 39: What is the midpoint of $A(-2,5)$ and $B(4,1)$?

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

Flashcard 40: What condition shows that vectors $\vec{u}$ and $\vec{v}$ are parallel (in $2$D)?

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

Flashcard 41: What is the vector projection of $\textbf{a}$ onto $\textbf{b}$?

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

Flashcard 42: What is the component form of the vector from $A(x_1,y_1)$ to $B(x_2,y_2)$?

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

Flashcard 43: Find the dot product of $\begin{bmatrix} 2 \ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \ 1 \end{bmatrix}$.

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

Flashcard 44: State the formula to find the angle $\theta$ between vectors $\mathbf{a}$ and $\mathbf{b}$.

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

Flashcard 45: What does a zero vector's magnitude equal to?

Answer:

1. Zero vector has no length by definition.

Flashcard 46: What is the component form of the vector from $A(x_1,y_1)$ to $B(x_2,y_2)$?

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

Flashcard 47: For vectors $\mathbf{a}$ and $\mathbf{b}$, which property is $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$?

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

Flashcard 48: If $\mathbf{u} = \langle 3, 4 \rangle$, what is $\|\mathbf{u}\|$?

Answer:

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

Flashcard 49: For vectors $\mathbf{a}$ and $\mathbf{b}$, which property is $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$?

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

Flashcard 50: Identify the operation used to find a vector's projection onto another.

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

Flashcard 51: Calculate the cross product of $\begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$.

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

Flashcard 52: Determine if vectors $\begin{bmatrix} 3 \ 6 \end{bmatrix}$ and $\begin{bmatrix} 2 \ 4 \end{bmatrix}$ are parallel.

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

Flashcard 53: Calculate the dot product of $\mathbf{a} = \langle 1, 0 \rangle$ and $\mathbf{b} = \langle 0, 1 \rangle$.

Answer:

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

Flashcard 54: What is the zero vector in $2$D in component form?

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

Flashcard 55: Calculate the magnitude of vector \begin{bmatrix} 5 \ 12 \\ \text{endbmatrix}.

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

Flashcard 56: What does a zero vector's magnitude equal to?

Answer:

1. Zero vector has no length by definition.

Flashcard 57: What is the vector from $A(1,2)$ to $B(6,-1)$ in component form?

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

Flashcard 58: What is the magnitude of $\mathbf{v} = \langle 6, 8 \rangle$?

Answer:

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

Flashcard 59: What is the magnitude of $\langle 3,4\rangle$?

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

Flashcard 60: What is the dot product of vectors $\textbf{u} = \begin{bmatrix} a \ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \ d \end{bmatrix}$?

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

Flashcard 61: What is the magnitude of the zero vector?

Answer: Zero. The zero vector has no length by definition.

Flashcard 62: Find the scalar multiple of $\mathbf{v} = \langle 2, 3 \rangle$ by $-2$.

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

Flashcard 63: Identify the zero vector in two dimensions.

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

Flashcard 64: Determine if vectors $\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}$ are orthogonal.

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

Flashcard 65: What is the vector addition result of $\mathbf{u} = \langle 0, 1 \rangle$ and $\mathbf{v} = \langle 1, 0 \rangle$?

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

Flashcard 66: What operation finds the angle between $\mathbf{a}$ and $\mathbf{b}$ if they are not zero vectors?

Answer: Dot product. Cosine formula requires dot product calculation.

Flashcard 67: What is a direction vector for the line through $A(x_1,y_1)$ and $B(x_2,y_2)$?

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

Flashcard 68: What is the direction of the vector $\mathbf{v} = \langle 0, 1 \rangle$?

Answer: Positive y-axis. Unit vector pointing upward.

Flashcard 69: What is the length of the vector $\mathbf{v} = \langle 0, 0, 0 \rangle$?

Answer:

1. Zero vector has zero magnitude.

Flashcard 70: If $\mathbf{v} = \langle 2, 3 \rangle$, what is $3\mathbf{v}$?

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

Flashcard 71: Identify a property of dot product concerning vector orthogonality.

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

Flashcard 72: What is the midpoint of segment $AB$ where $A(x_1,y_1)$ and $B(x_2,y_2)$?

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

Flashcard 73: Find the scalar multiple of $\mathbf{v} = \langle 2, 3 \rangle$ by $-2$.

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

Flashcard 74: Identify the operation used to find a vector's projection onto another.

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

Flashcard 75: What is the unit vector in the direction of $\mathbf{v} = \langle a, b \rangle$?

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

Flashcard 76: What is the angle between vectors $\begin{bmatrix} 1 \ 0 \ \end{bmatrix}$ and $\begin{bmatrix} 0 \ 1 \ \end{bmatrix}$?

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

Flashcard 77: Identify the scalar multiplication property for vector $\textbf{v}$ and scalar $0$.

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

Flashcard 78: What is a perpendicular direction vector to $\langle 3,-5\rangle$?

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

Flashcard 79: What is the result of scalar multiplication $k\langle a,b\rangle$?

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

Flashcard 80: How do you calculate the direction angle of vector $\textbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$?

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

Flashcard 81: If $\mathbf{v} = \langle 2, 3 \rangle$, what is $3\mathbf{v}$?

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

Flashcard 82: What is the midpoint of segment $AB$ where $A(x_1,y_1)$ and $B(x_2,y_2)$?

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

Flashcard 83: What is the slope of the line with direction vector $\langle 6,-4\rangle$?

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

Flashcard 84: What is the result of subtracting $\langle a,b\rangle-\langle c,d\rangle$?

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

Flashcard 85: What is the result of scalar multiplication $k\langle a,b\rangle$?

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

Flashcard 86: Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \ 4 \end{bmatrix}$

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

Flashcard 87: What is the component form of $\overrightarrow{PQ}$ for $P(-1,4)$ and $Q(2,0)$?

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

Flashcard 88: What is the result of adding $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle 3, 4 \rangle$?

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

Flashcard 89: State the result of the cross product of parallel vectors.

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

Flashcard 90: What is the formula for the dot product of vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$?

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

Flashcard 91: Determine if vectors $\begin{bmatrix} 3 \ 6 \ \end{bmatrix}$ and $\begin{bmatrix} 2 \ 4 \ \end{bmatrix}$ are parallel.

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

Flashcard 92: Calculate the magnitude of $\mathbf{v} = \langle 1, 2, 2 \rangle$.

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

Flashcard 93: What is the direction of the vector $\mathbf{v} = \langle 0, 1 \rangle$?

Answer: Positive y-axis. Unit vector pointing upward.

Flashcard 94: State the distributive property for vectors and scalar multiplication.

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

Flashcard 95: What is the formula for the dot product of vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$?

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

Flashcard 96: What is $-3\langle 4,-2\rangle$?

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

Flashcard 97: What is the distance between points $A(x_1,y_1)$ and $B(x_2,y_2)$ using vectors?

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

Flashcard 98: What is the magnitude of the vector $\vec{v}=\langle a,b\rangle$ in the plane?

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

Flashcard 99: What is the zero vector in 3D space?

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

Flashcard 100: What is the result of adding $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle 3, 4 \rangle$?

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