AP Precalculus 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 AP Precalculus.

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 definition of a vector?

Answer: A vector is a quantity with both magnitude and direction. This distinguishes vectors from scalars which only have magnitude.

Flashcard 2: How is a vector typically represented in a plane?

Answer: As an arrow from an initial point to a terminal point. The arrow shows both direction and magnitude visually.

Flashcard 3: What is meant by the term 'orthogonal vectors'?

Answer: Vectors with a dot product of zero. Perpendicular vectors meet at right angles.

Flashcard 4: What is meant by the term 'orthogonal vectors'?

Answer: Vectors with a dot product of zero. Perpendicular vectors meet at right angles.

Flashcard 5: Find the projection of \begin{bmatrix} 3 \ 4 \matrix} onto \begin{bmatrix} 6 \ 8 \matrix}.

Answer: \begin{bmatrix} 3 \ 4 \matrix}. Both vectors are parallel, so projection equals the first vector.

Flashcard 6: Find the sum of vectors $\begin{bmatrix} -2 \ 5 \end{bmatrix}$ and $\begin{bmatrix} 7 \ -3 \end{bmatrix}$.

Answer: $\begin{bmatrix} 5 \ 2 \end{bmatrix}$. Add components: $(-2+7, 5-3) = (5,2)$.

Flashcard 7: Find the vector sum of $\textbf{a} = \begin{bmatrix} 3 \ -2 \end{bmatrix}$ and $\textbf{b} = \begin{bmatrix} -1 \ 4 \end{bmatrix}$.

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

Flashcard 8: What operation is used to find a vector's component along another vector?

Answer: Projection. Projects one vector onto the direction of another.

Flashcard 9: Find the magnitude of \begin{bmatrix} 7 \ 24 \matrix}.

Answer: $25$. Use $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25$.

Flashcard 10: What is the definition of a vector?

Answer: A vector is a quantity with both magnitude and direction. This distinguishes vectors from scalars which only have magnitude.

Flashcard 11: State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.

Answer: $\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.

Flashcard 12: State the formula for the dot product of \textbf{u} = \begin{bmatrix} a \ b \matrix} and \textbf{v} = \begin{bmatrix} c \ d \matrix}.

Answer: $a \times c + b \times d$. Multiply corresponding components and sum the results.

Flashcard 13: Find the angle between $\begin{bmatrix} 1 \ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \ 1 \end{bmatrix}$.

Answer: $90^\circ$. Standard unit vectors are perpendicular to each other.

Flashcard 14: Find the sum of vectors \begin{bmatrix} -2 \ 5 \matrix} and \begin{bmatrix} 7 \ -3 \matrix}.

Answer: \begin{bmatrix} 5 \ 2 \matrix}. Add components: $(-2+7, 5-3) = (5,2)$.

Flashcard 15: Find the scalar projection of \begin{bmatrix} 3 \ 4 \matrix} on \begin{bmatrix} 4 \ 3 \matrix}.

Answer: $\frac{24}{5}$. Scalar projection: $\frac{\textbf{u} \cdot \textbf{v}}{|\textbf{v}|} = \frac{24}{5}$.

Flashcard 16: What does a negative scalar multiplication do to a vector?

Answer: Reverses the direction of the vector. Changes direction but preserves magnitude.

Flashcard 17: State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.

Answer: $\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.

Flashcard 18: Find the magnitude of \begin{bmatrix} 7 \ 24 \matrix}.

Answer: $25$. Use $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25$.

Flashcard 19: What is the direction of a vector if its components are all equal?

Answer: The direction is along the line $y=x=z$. Equal components create a vector along the main diagonal.

Flashcard 20: Find the angle between \begin{bmatrix} 1 \ 0 \matrix} and \begin{bmatrix} 0 \ 1 \matrix}.

Answer: $90^\text{°}$. Standard unit vectors are perpendicular to each other.

Flashcard 21: Find the magnitude of vector \begin{bmatrix} 3 \ 4 \matrix}.

Answer: $5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

Flashcard 22: What is the magnitude of the zero vector?

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

Flashcard 23: State the associative property of vector addition.

Answer: ($\textbf{u} + \textbf{v}) + \textbf{w} = \textbf{u} + (\textbf{v} + \textbf{w})$. Grouping doesn't affect vector addition results.

Flashcard 24: State the formula for the dot product of $\textbf{u} = \begin{bmatrix} a \ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \ d \end{bmatrix}$.

Answer: $a \times c + b \times d$. Multiply corresponding components and sum the results.

Flashcard 25: Find the dot product of \begin{bmatrix} 1 \ 3 \matrix} and \begin{bmatrix} 4 \ -2 \matrix}.

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

Flashcard 26: How do you find the angle $\theta$ between two vectors $\textbf{u}$ and $\textbf{v}$?

Answer: $\theta = \text{acos}\bigg(\frac{\textbf{u} \bullet \textbf{v}}{|\textbf{u}||\textbf{v}|}\bigg)$. Uses the dot product formula and inverse cosine function.

Flashcard 27: Find the magnitude of vector $\begin{bmatrix} 3 \ 4 \end{bmatrix}$

Answer: $5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$

Flashcard 28: What is the inverse of vector $\textbf{v}$?

Answer: The vector $-\textbf{v}$. (Negation). The additive inverse of a vector.

Flashcard 29: What operation is used to find a vector's component along another vector?

Answer: Projection. Projects one vector onto the direction of another.

Flashcard 30: What is the result of multiplying a vector by a scalar $k$?

Answer: A vector in the same direction if $k > 0$, opposite if $k < 0$. Scales magnitude and may reverse direction based on sign.

Flashcard 31: Identify the property: $\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$.

Answer: Commutative property of vector addition. Vector addition is commutative like regular addition.

Flashcard 32: What is the result of scalar multiplication of a vector by zero?

Answer: A zero vector. Multiplying by zero eliminates all magnitude and direction.

Flashcard 33: What is the magnitude of \begin{bmatrix} 0 \ 0 \ 0 \matrix}?

Answer: Zero. The zero vector has zero magnitude by definition.

Flashcard 34: Determine if vectors \begin{bmatrix} 2 \ 3 \matrix} and \begin{bmatrix} -4 \ -6 \matrix} are parallel.

Answer: Yes, they are parallel. The second vector is $-2$ times the first vector.

Flashcard 35: State the associative property of vector addition.

Answer: ($\textbf{u} + \textbf{v}) + \textbf{w} = \textbf{u} + (\textbf{v} + \textbf{w})$. Grouping doesn't affect vector addition results.

Flashcard 36: Find the unit vector of \textbf{v} = \begin{bmatrix} 5 \ 12 \matrix}.

Answer: \begin{bmatrix} \frac{5}{13} \ \frac{12}{13} \matrix}. Divide by magnitude $13$ to get unit vector.

Flashcard 37: What does a negative scalar multiplication do to a vector?

Answer: Reverses the direction of the vector. Changes direction but preserves magnitude.

Flashcard 38: What is the inverse of vector $\textbf{v}$?

Answer: The vector $-\textbf{v}$. (Negation). The additive inverse of a vector.

Flashcard 39: What is the direction of a vector if its components are all equal?

Answer: The direction is along the line $y=x=z$. Equal components create a vector along the main diagonal.

Flashcard 40: What does it mean if the dot product of two vectors is zero?

Answer: The vectors are orthogonal (perpendicular). Zero dot product indicates 90° angle between vectors.

Flashcard 41: Find the projection of $\begin{bmatrix} 3 \ 4 \end{bmatrix}$ onto $\begin{bmatrix} 6 \ 8 \end{bmatrix}$.

Answer: $\begin{bmatrix} 3 \ 4 \end{bmatrix}$. Both vectors are parallel, so projection equals the first vector.

Flashcard 42: What is the result of multiplying a vector by a scalar $k$?

Answer: A vector in the same direction if $k > 0$, opposite if $k < 0$. Scales magnitude and may reverse direction based on sign.

Flashcard 43: How do you find the angle $\theta$ between two vectors $\textbf{u}$ and $\textbf{v}$?

Answer: $\theta = \text{acos}\bigg(\frac{\textbf{u} \bullet \textbf{v}}{|\textbf{u}||\textbf{v}|}\bigg)$. Uses the dot product formula and inverse cosine function.

Flashcard 44: What is a linear combination of vectors?

Answer: A sum of scalar multiples of vectors. Combines vectors with scalar coefficients.

Flashcard 45: What does it mean for vectors to be linearly dependent?

Answer: One vector is a linear combination of the others. One vector can be expressed using the others.

Flashcard 46: Find the vector sum of \textbf{a} = \begin{bmatrix} 3 \ -2 \matrix} and \textbf{b} = \begin{bmatrix} -1 \ 4 \matrix}.

Answer: \begin{bmatrix} 2 \ 2 \matrix}. Add corresponding components: $(3-1, -2+4) = (2,2)$.

Flashcard 47: What is the magnitude of $\begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}$?

Answer: Zero. The zero vector has zero magnitude by definition.

Flashcard 48: Find the dot product of $\begin{bmatrix} 1 \ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \ -2 \end{bmatrix}$.

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

Flashcard 49: Which condition indicates that two vectors are parallel?

Answer: One vector is a scalar multiple of the other. Parallel vectors point in the same or opposite directions.

Flashcard 50: What is the result of adding a vector to its negative?

Answer: The zero vector. A vector plus its additive inverse equals zero vector.

Flashcard 51: What is a linear combination of vectors?

Answer: A sum of scalar multiples of vectors. Combines vectors with scalar coefficients.

Flashcard 52: What is the magnitude of the zero vector?

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

Flashcard 53: How is a vector typically represented in a plane?

Answer: As an arrow from an initial point to a terminal point. The arrow shows both direction and magnitude visually.

Flashcard 54: Determine if vectors $\begin{bmatrix} 2 \ 3 \ \end{bmatrix}$ and $\begin{bmatrix} -4 \ -6 \ \end{bmatrix}$ are parallel.

Answer: Yes, they are parallel. The second vector is $-2$ times the first vector.

Flashcard 55: What does it mean for vectors to be linearly dependent?

Answer: One vector is a linear combination of the others. One vector can be expressed using the others.