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

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

Flashcard 3: 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 4: 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 5: 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 6: What operation is used to find a vector's component along another vector?

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

Flashcard 7: 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 8: 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 9: 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 10: 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 11: 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 12: What does a negative scalar multiplication do to a vector?

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

Flashcard 13: Find the magnitude of \begin{bmatrix} 7 \ 24 \matrix}.

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

Flashcard 14: 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 15: 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 16: 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 17: What is the magnitude of the zero vector?

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

Flashcard 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: What is the magnitude of \begin{bmatrix} 0 \ 0 \ 0 \matrix}?

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

Flashcard 26: 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 27: 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 28: What is the inverse of vector $\textbf{v}$?

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

Flashcard 29: 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 30: 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 31: 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 32: What is a linear combination of vectors?

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

Flashcard 33: 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 34: 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 35: What is the magnitude of $\begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}$?

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

Flashcard 36: 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 37: 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 38: 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 39: 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 40: 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.