Magnitude and Direction of Scaled Vectors - Pre-Calculus
Card 1 of 30
Choose the correct statement: if $0<|c|<1$, then $|c\vec{v}|$ is what relative to $|\vec{v}|$?
Choose the correct statement: if $0<|c|<1$, then $|c\vec{v}|$ is what relative to $|\vec{v}|$?
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Smaller: $|c\vec{v}|<|\vec{v}|$ for $\vec{v}\neq\vec{0}$. Since $0<|c|<1$, we have $|c||\vec{v}|<|\vec{v}|$.
Smaller: $|c\vec{v}|<|\vec{v}|$ for $\vec{v}\neq\vec{0}$. Since $0<|c|<1$, we have $|c||\vec{v}|<|\vec{v}|$.
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State the formula for the magnitude of a scalar multiple $c\vec{v}$ in terms of $c$ and $|\vec{v}|$.
State the formula for the magnitude of a scalar multiple $c\vec{v}$ in terms of $c$ and $|\vec{v}|$.
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$|c\vec{v}|=|c|,|\vec{v}|$. The magnitude scales by the absolute value of the scalar.
$|c\vec{v}|=|c|,|\vec{v}|$. The magnitude scales by the absolute value of the scalar.
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What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c>0$ and $\vec{v}\neq\vec{0}$?
What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c>0$ and $\vec{v}\neq\vec{0}$?
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Along $\vec{v}$ (same direction). Positive scalars preserve the vector's direction.
Along $\vec{v}$ (same direction). Positive scalars preserve the vector's direction.
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What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c<0$ and $\vec{v}\neq\vec{0}$?
What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c<0$ and $\vec{v}\neq\vec{0}$?
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Against $\vec{v}$ (opposite direction). Negative scalars reverse the vector's direction.
Against $\vec{v}$ (opposite direction). Negative scalars reverse the vector's direction.
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What is $|c\vec{v}|$ when $c=0$ (for any vector $\vec{v}$)?
What is $|c\vec{v}|$ when $c=0$ (for any vector $\vec{v}$)?
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$|0\vec{v}|=0$. Zero scalar always produces zero vector with zero magnitude.
$|0\vec{v}|=0$. Zero scalar always produces zero vector with zero magnitude.
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Identify the direction of $(-1)\vec{v}$ relative to $\vec{v}$ for $\vec{v}\neq\vec{0}$.
Identify the direction of $(-1)\vec{v}$ relative to $\vec{v}$ for $\vec{v}\neq\vec{0}$.
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Opposite direction to $\vec{v}$. Multiplying by $-1$ reverses direction while preserving magnitude.
Opposite direction to $\vec{v}$. Multiplying by $-1$ reverses direction while preserving magnitude.
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Identify the magnitude of $(-1)\vec{v}$ in terms of $|\vec{v}|$.
Identify the magnitude of $(-1)\vec{v}$ in terms of $|\vec{v}|$.
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$|(-1)\vec{v}|=|\vec{v}|$. Since $|-1|=1$, the magnitude remains unchanged.
$|(-1)\vec{v}|=|\vec{v}|$. Since $|-1|=1$, the magnitude remains unchanged.
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What is $|2\vec{v}|$ in terms of $|\vec{v}|$?
What is $|2\vec{v}|$ in terms of $|\vec{v}|$?
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$|2\vec{v}|=2|\vec{v}|$. Apply $|c\vec{v}|=|c||\vec{v}|$ with $|2|=2$.
$|2\vec{v}|=2|\vec{v}|$. Apply $|c\vec{v}|=|c||\vec{v}|$ with $|2|=2$.
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What is $|\frac{1}{3}\vec{v}|$ in terms of $|\vec{v}|$?
What is $|\frac{1}{3}\vec{v}|$ in terms of $|\vec{v}|$?
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$|\frac{1}{3}\vec{v}|=\frac{1}{3}|\vec{v}|$. Apply $|c\vec{v}|=|c||\vec{v}|$ with $|\frac{1}{3}|=\frac{1}{3}$.
$|\frac{1}{3}\vec{v}|=\frac{1}{3}|\vec{v}|$. Apply $|c\vec{v}|=|c||\vec{v}|$ with $|\frac{1}{3}|=\frac{1}{3}$.
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Find $|(-4)\vec{v}|$ if $|\vec{v}|=7$.
Find $|(-4)\vec{v}|$ if $|\vec{v}|=7$.
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$|(-4)\vec{v}|=28$. Use $|c\vec{v}|=|c||\vec{v}|=|-4|\cdot 7=4\cdot 7$.
$|(-4)\vec{v}|=28$. Use $|c\vec{v}|=|c||\vec{v}|=|-4|\cdot 7=4\cdot 7$.
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Find $|3\vec{v}|$ if $|\vec{v}|=\frac{5}{2}$.
Find $|3\vec{v}|$ if $|\vec{v}|=\frac{5}{2}$.
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$|3\vec{v}|=\frac{15}{2}$. Use $|c\vec{v}|=|c||\vec{v}|=3\cdot\frac{5}{2}$.
$|3\vec{v}|=\frac{15}{2}$. Use $|c\vec{v}|=|c||\vec{v}|=3\cdot\frac{5}{2}$.
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Find $|\frac{-1}{2}\vec{v}|$ if $|\vec{v}|=12$.
Find $|\frac{-1}{2}\vec{v}|$ if $|\vec{v}|=12$.
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$|\frac{-1}{2}\vec{v}|=6$. Use $|c\vec{v}|=|c||\vec{v}|=\frac{1}{2}\cdot 12$.
$|\frac{-1}{2}\vec{v}|=6$. Use $|c\vec{v}|=|c||\vec{v}|=\frac{1}{2}\cdot 12$.
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Identify the direction of $(-5)\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.
Identify the direction of $(-5)\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.
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Opposite direction to $\vec{v}$. Since $-5<0$, the direction reverses.
Opposite direction to $\vec{v}$. Since $-5<0$, the direction reverses.
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Identify the direction of $\frac{7}{3}\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.
Identify the direction of $\frac{7}{3}\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.
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Same direction as $\vec{v}$. Since $\frac{7}{3}>0$, the direction is preserved.
Same direction as $\vec{v}$. Since $\frac{7}{3}>0$, the direction is preserved.
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What is the magnitude scale factor from $\vec{v}$ to $c\vec{v}$ (assume $\vec{v}\neq\vec{0}$)?
What is the magnitude scale factor from $\vec{v}$ to $c\vec{v}$ (assume $\vec{v}\neq\vec{0}$)?
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Multiply by $|c|$. The formula $|c\vec{v}|=|c||\vec{v}|$ shows magnitude scales by $|c|$.
Multiply by $|c|$. The formula $|c\vec{v}|=|c||\vec{v}|$ shows magnitude scales by $|c|$.
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Find $c$ if $|\vec{v}|=9$, $c>0$, and $|c\vec{v}|=27$.
Find $c$ if $|\vec{v}|=9$, $c>0$, and $|c\vec{v}|=27$.
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$c=3$. Solve $27=|c|\cdot 9$ with $c>0$, so $|c|=c=3$.
$c=3$. Solve $27=|c|\cdot 9$ with $c>0$, so $|c|=c=3$.
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Find $|c|$ if $|\vec{v}|=8$ and $|c\vec{v}|=2$.
Find $|c|$ if $|\vec{v}|=8$ and $|c\vec{v}|=2$.
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$|c|=\frac{1}{4}$. From $|c\vec{v}|=|c||\vec{v}|$: $2=|c|\cdot 8$.
$|c|=\frac{1}{4}$. From $|c\vec{v}|=|c||\vec{v}|$: $2=|c|\cdot 8$.
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Find $|c\vec{v}|$ if $c=-0.2$ and $|\vec{v}|=30$.
Find $|c\vec{v}|$ if $c=-0.2$ and $|\vec{v}|=30$.
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$|c\vec{v}|=6$. Calculate $|c\vec{v}|=|-0.2|\cdot 30=0.2\cdot 30$.
$|c\vec{v}|=6$. Calculate $|c\vec{v}|=|-0.2|\cdot 30=0.2\cdot 30$.
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Identify when $c\vec{v}$ has the same direction as $\vec{v}$ (assume $\vec{v}\neq\vec{0}$).
Identify when $c\vec{v}$ has the same direction as $\vec{v}$ (assume $\vec{v}\neq\vec{0}$).
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When $c>0$. Positive scalars maintain the original vector direction.
When $c>0$. Positive scalars maintain the original vector direction.
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Find $|\frac{1}{2}\mathbf{v}|$ if $|\mathbf{v}|=18$.
Find $|\frac{1}{2}\mathbf{v}|$ if $|\mathbf{v}|=18$.
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$9$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $\frac{1}{2} \cdot 18 = 9$.
$9$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $\frac{1}{2} \cdot 18 = 9$.
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If $\mathbf{v}\neq\mathbf{0}$ and $c\mathbf{v}$ points opposite $\mathbf{v}$, which sign must $c$ have?
If $\mathbf{v}\neq\mathbf{0}$ and $c\mathbf{v}$ points opposite $\mathbf{v}$, which sign must $c$ have?
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$c<0$. Vectors point oppositely only when the scalar is negative.
$c<0$. Vectors point oppositely only when the scalar is negative.
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If $|c\mathbf{v}|=0$ and $|\mathbf{v}|\neq 0$, what must $c$ equal?
If $|c\mathbf{v}|=0$ and $|\mathbf{v}|\neq 0$, what must $c$ equal?
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$c=0$. Only $c=0$ makes $|c\mathbf{v}|=0$ when $\mathbf{v} \neq \mathbf{0}$.
$c=0$. Only $c=0$ makes $|c\mathbf{v}|=0$ when $\mathbf{v} \neq \mathbf{0}$.
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If $|c\mathbf{v}|=3$ and $|\mathbf{v}|=12$, what is $|c|$?
If $|c\mathbf{v}|=3$ and $|\mathbf{v}|=12$, what is $|c|$?
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$|c|=\frac{1}{4}$. From $|c\mathbf{v}|=|c||\mathbf{v}|$: $3 = |c| \cdot 12$, so $|c| = \frac{1}{4}$.
$|c|=\frac{1}{4}$. From $|c\mathbf{v}|=|c||\mathbf{v}|$: $3 = |c| \cdot 12$, so $|c| = \frac{1}{4}$.
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If $|c\mathbf{v}|=24$ and $|\mathbf{v}|=6$, what is $|c|$?
If $|c\mathbf{v}|=24$ and $|\mathbf{v}|=6$, what is $|c|$?
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$|c|=4$. From $|c\mathbf{v}|=|c||\mathbf{v}|$: $24 = |c| \cdot 6$, so $|c| = 4$.
$|c|=4$. From $|c\mathbf{v}|=|c||\mathbf{v}|$: $24 = |c| \cdot 6$, so $|c| = 4$.
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Compute $|(-2)\mathbf{v}|$ in terms of $|\mathbf{v}|$.
Compute $|(-2)\mathbf{v}|$ in terms of $|\mathbf{v}|$.
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$|(-2)\mathbf{v}|=2|\mathbf{v}|$. Since $|-2| = 2$, the magnitude doubles.
$|(-2)\mathbf{v}|=2|\mathbf{v}|$. Since $|-2| = 2$, the magnitude doubles.
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What is the direction of $0.2\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?
What is the direction of $0.2\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?
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Same direction as $\mathbf{v}$. Since $0.2 > 0$, the vector maintains its direction.
Same direction as $\mathbf{v}$. Since $0.2 > 0$, the vector maintains its direction.
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What is the direction of $-7\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?
What is the direction of $-7\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?
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Opposite direction to $\mathbf{v}$. Since $-7 < 0$, the vector points oppositely.
Opposite direction to $\mathbf{v}$. Since $-7 < 0$, the vector points oppositely.
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Find $|-\frac{3}{5}\mathbf{v}|$ if $|\mathbf{v}|=20$.
Find $|-\frac{3}{5}\mathbf{v}|$ if $|\mathbf{v}|=20$.
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$12$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $\frac{3}{5} \cdot 20 = 12$.
$12$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $\frac{3}{5} \cdot 20 = 12$.
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Find $|c\mathbf{v}|$ if $c=-0.3$ and $|\mathbf{v}|=50$.
Find $|c\mathbf{v}|$ if $c=-0.3$ and $|\mathbf{v}|=50$.
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$15$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $|-0.3| \cdot 50 = 0.3 \cdot 50 = 15$.
$15$. Apply $|c\mathbf{v}|=|c||\mathbf{v}|$: $|-0.3| \cdot 50 = 0.3 \cdot 50 = 15$.
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State the magnitude rule for a scalar multiple $c\mathbf{v}$ in terms of $|\mathbf{v}|$.
State the magnitude rule for a scalar multiple $c\mathbf{v}$ in terms of $|\mathbf{v}|$.
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$|c\mathbf{v}|=|c|,|\mathbf{v}|$. The magnitude of a scalar multiple equals the absolute value of the scalar times the original magnitude.
$|c\mathbf{v}|=|c|,|\mathbf{v}|$. The magnitude of a scalar multiple equals the absolute value of the scalar times the original magnitude.
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