Precalculus Flashcards: Magnitude And Direction Of Scaled Vectors

What this deck covers

This deck focuses on Magnitude And Direction Of Scaled Vectors, giving you a quick way to review the definitions, rules, and examples that matter most for 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: Find $\|3\vec{v}\|$ if $\|\vec{v}\|=\frac{5}{2}$.

Answer: $\|3\vec{v}\|=\frac{15}{2}$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=3\cdot\frac{5}{2}$.

Flashcard 2: What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c<0$ and $\vec{v}\neq\vec{0}$?

Answer: Against $\vec{v}$ (opposite direction). Negative scalars reverse the vector's direction.

Flashcard 3: Find $\|\frac{1}{2}\mathbf{v}\|$ if $\|\mathbf{v}\|=18$.

Answer: $9$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $\frac{1}{2} \cdot 18 = 9$.

Flashcard 4: If $\|c\mathbf{v}\|=24$ and $\|\mathbf{v}\|=6$, what is $|c|$?

Answer: $|c|=4$. From $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $24 = |c| \cdot 6$, so $|c| = 4$.

Flashcard 5: Find $\|(-4)\vec{v}\|$ if $\|\vec{v}\|=7$.

Answer: $\|(-4)\vec{v}\|=28$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=|-4|\cdot 7=4\cdot 7$.

Flashcard 6: What is the direction of $-7\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?

Answer: Opposite direction to $\mathbf{v}$. Since $-7 < 0$, the vector points oppositely.

Flashcard 7: Find $\|\frac{-1}{2}\vec{v}\|$ if $\|\vec{v}\|=12$.

Answer: $\|\frac{-1}{2}\vec{v}\|=6$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=\frac{1}{2}\cdot 12$.

Flashcard 8: What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c>0$ and $\vec{v}\neq\vec{0}$?

Answer: Along $\vec{v}$ (same direction). Positive scalars preserve the vector's direction.

Flashcard 9: Choose the correct statement: if $0<|c|<1$, then $\|c\vec{v}\|$ is what relative to $\|\vec{v}\|$?

Answer: Smaller: $\|c\vec{v}\|<\|\vec{v}\|$ for $\vec{v}\neq\vec{0}$. Since $0<|c|<1$, we have $|c|\|\vec{v}\|<\|\vec{v}\|$.

Flashcard 10: What is the direction of $0.2\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?

Answer: Same direction as $\mathbf{v}$. Since $0.2 > 0$, the vector maintains its direction.

Flashcard 11: Identify the scalar $c$ that keeps $c\mathbf{v}$ pointing in the same direction as $\mathbf{v}$ (assuming $\mathbf{v}\neq\mathbf{0}$).

Answer: Any $c>0$. Positive scalars maintain the original direction.

Flashcard 12: Find $\|-4\mathbf{v}\|$ if $\|\mathbf{v}\|=2.5$.

Answer: $10$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|-4| \cdot 2.5 = 10$.

Flashcard 13: State the formula for the magnitude of a scalar multiple $c\vec{v}$ in terms of $c$ and $\|\vec{v}\|$.

Answer: $\|c\vec{v}\|=|c|\,\|\vec{v}\|$. The magnitude scales by the absolute value of the scalar.

Flashcard 14: State the magnitude rule for a scalar multiple $c\mathbf{v}$ in terms of $\|\mathbf{v}\|$.

Answer: $\|c\mathbf{v}\|=|c|\,\|\mathbf{v}\|$. The magnitude of a scalar multiple equals the absolute value of the scalar times the original magnitude.

Flashcard 15: Find $\|3\mathbf{v}\|$ if $\|\mathbf{v}\|=5$.

Answer: $15$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|3| \cdot 5 = 15$.

Flashcard 16: Find $\|c\vec{v}\|$ if $c=-0.2$ and $\|\vec{v}\|=30$.

Answer: $\|c\vec{v}\|=6$. Calculate $\|c\vec{v}\|=|-0.2|\cdot 30=0.2\cdot 30$.

Flashcard 17: What is the direction of $c\mathbf{v}$ relative to $\mathbf{v}$ when $c>0$ and $\mathbf{v}\neq \mathbf{0}$?

Answer: Along $\mathbf{v}$ (same direction). Positive scalars preserve the vector's direction.

Flashcard 18: Compute $\|(-2)\mathbf{v}\|$ in terms of $\|\mathbf{v}\|$.

Answer: $\|(-2)\mathbf{v}\|=2\|\mathbf{v}\|$. Since $|-2| = 2$, the magnitude doubles.

Flashcard 19: What is $\|c\mathbf{v}\|$ when $c=0$ (for any vector $\mathbf{v}$)?

Answer: $\|0\mathbf{v}\|=0$. Zero scalar always produces zero magnitude regardless of the vector.

Flashcard 20: What is $\|\frac{1}{3}\vec{v}\|$ in terms of $\|\vec{v}\|$?

Answer: $\|\frac{1}{3}\vec{v}\|=\frac{1}{3}\|\vec{v}\|$. Apply $\|c\vec{v}\|=|c|\|\vec{v}\|$ with $|\frac{1}{3}|=\frac{1}{3}$.

Flashcard 21: Find $\|c\mathbf{v}\|$ if $c=1.6$ and $\|\mathbf{v}\|=10$.

Answer: $16$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|1.6| \cdot 10 = 1.6 \cdot 10 = 16$.

Flashcard 22: Find $|c|$ if $\|\vec{v}\|=8$ and $\|c\vec{v}\|=2$.

Answer: $|c|=\frac{1}{4}$. From $\|c\vec{v}\|=|c|\|\vec{v}\|$: $2=|c|\cdot 8$.

Flashcard 23: Identify the direction of $(-5)\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.

Answer: Opposite direction to $\vec{v}$. Since $-5<0$, the direction reverses.

Flashcard 24: What is the direction of $c\mathbf{v}$ relative to $\mathbf{v}$ when $c<0$ and $\mathbf{v}\neq \mathbf{0}$?

Answer: Against $\mathbf{v}$ (opposite direction). Negative scalars reverse the vector's direction.

Flashcard 25: Find $\|-\frac{3}{5}\mathbf{v}\|$ if $\|\mathbf{v}\|=20$.

Answer: $12$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $\frac{3}{5} \cdot 20 = 12$.

Flashcard 26: Find $c$ if $\|\vec{v}\|=9$, $c>0$, and $\|c\vec{v}\|=27$.

Answer: $c=3$. Solve $27=|c|\cdot 9$ with $c>0$, so $|c|=c=3$.

Flashcard 27: Identify when $c\vec{v}$ has the same direction as $\vec{v}$ (assume $\vec{v}\neq\vec{0}$).

Answer: When $c>0$. Positive scalars maintain the original vector direction.

Flashcard 28: Identify the scalar $c$ that reverses the direction of $\mathbf{v}$ in $c\mathbf{v}$ (assuming $\mathbf{v}\neq\mathbf{0}$).

Answer: Any $c<0$. Negative scalars flip the vector to point oppositely.

Flashcard 29: What is the magnitude scale factor from $\vec{v}$ to $c\vec{v}$ (assume $\vec{v}\neq\vec{0}$)?

Answer: Multiply by $|c|$. The formula $\|c\vec{v}\|=|c|\|\vec{v}\|$ shows magnitude scales by $|c|$.

Flashcard 30: Identify the direction of $\frac{7}{3}\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.

Answer: Same direction as $\vec{v}$. Since $\frac{7}{3}>0$, the direction is preserved.

Flashcard 31: If $\|c\mathbf{v}\|=3$ and $\|\mathbf{v}\|=12$, what is $|c|$?

Answer: $|c|=\frac{1}{4}$. From $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $3 = |c| \cdot 12$, so $|c| = \frac{1}{4}$.

Flashcard 32: Identify the direction of $(-1)\vec{v}$ relative to $\vec{v}$ for $\vec{v}\neq\vec{0}$.

Answer: Opposite direction to $\vec{v}$. Multiplying by $-1$ reverses direction while preserving magnitude.

Flashcard 33: What is $\|2\vec{v}\|$ in terms of $\|\vec{v}\|$?

Answer: $\|2\vec{v}\|=2\|\vec{v}\|$. Apply $\|c\vec{v}\|=|c|\|\vec{v}\|$ with $|2|=2$.

Flashcard 34: What is $\|c\vec{v}\|$ when $c=0$ (for any vector $\vec{v}$)?

Answer: $\|0\vec{v}\|=0$. Zero scalar always produces zero vector with zero magnitude.

Flashcard 35: Find $\|c\mathbf{v}\|$ if $c=-0.3$ and $\|\mathbf{v}\|=50$.

Answer: $15$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|-0.3| \cdot 50 = 0.3 \cdot 50 = 15$.

Flashcard 36: Compute $\|(-1)\mathbf{v}\|$ in terms of $\|\mathbf{v}\|$.

Answer: $\|(-1)\mathbf{v}\|=\|\mathbf{v}\|$. Since $|-1| = 1$, the magnitude remains unchanged.

Flashcard 37: Identify the magnitude of $(-1)\vec{v}$ in terms of $\|\vec{v}\|$.

Answer: $\|(-1)\vec{v}\|=\|\vec{v}\|$. Since $|-1|=1$, the magnitude remains unchanged.

Flashcard 38: If $\mathbf{v}\neq\mathbf{0}$ and $c\mathbf{v}$ points opposite $\mathbf{v}$, which sign must $c$ have?

Answer: $c<0$. Vectors point oppositely only when the scalar is negative.

Flashcard 39: If $\|c\mathbf{v}\|=0$ and $\|\mathbf{v}\|\neq 0$, what must $c$ equal?

Answer: $c=0$. Only $c=0$ makes $\|c\mathbf{v}\|=0$ when $\mathbf{v} \neq \mathbf{0}$.