Show Scalar Multiplication Visually - Pre-Calculus
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What is the scalar $c$ if $c(3,-2)=(12,-8)$?
What is the scalar $c$ if $c(3,-2)=(12,-8)$?
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$c=4$. Since $4(3)=12$ and $4(-2)=-8$, $c=4$.
$c=4$. Since $4(3)=12$ and $4(-2)=-8$, $c=4$.
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Identify the correct statement: If $c=-5$, how does $c\vec{v}$ compare to $\vec{v}$?
Identify the correct statement: If $c=-5$, how does $c\vec{v}$ compare to $\vec{v}$?
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Opposite direction and $5$ times as long. Negative scalar reverses direction; $|-5|=5$ scales magnitude.
Opposite direction and $5$ times as long. Negative scalar reverses direction; $|-5|=5$ scales magnitude.
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What is the component-wise rule for scalar multiplication $c(v_x, v_y)$?
What is the component-wise rule for scalar multiplication $c(v_x, v_y)$?
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$c(v_x, v_y) = (cv_x, cv_y)$. Multiply each component by the scalar $c$.
$c(v_x, v_y) = (cv_x, cv_y)$. Multiply each component by the scalar $c$.
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What happens to a vector’s direction when it is multiplied by a scalar $c>0$?
What happens to a vector’s direction when it is multiplied by a scalar $c>0$?
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Direction stays the same. Positive scalars preserve the vector's direction.
Direction stays the same. Positive scalars preserve the vector's direction.
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What happens to a vector’s direction when it is multiplied by a scalar $c<0$?
What happens to a vector’s direction when it is multiplied by a scalar $c<0$?
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Direction reverses. Negative scalars flip the vector to point opposite.
Direction reverses. Negative scalars flip the vector to point opposite.
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What happens to a vector’s magnitude when it is multiplied by a scalar $c$?
What happens to a vector’s magnitude when it is multiplied by a scalar $c$?
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Magnitude is multiplied by $|c|$. The scalar's absolute value scales the length.
Magnitude is multiplied by $|c|$. The scalar's absolute value scales the length.
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What is the scalar multiple $3(2,-5)$?
What is the scalar multiple $3(2,-5)$?
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$(6,-15)$. Multiply each component: $3(2)=6$, $3(-5)=-15$.
$(6,-15)$. Multiply each component: $3(2)=6$, $3(-5)=-15$.
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What is the scalar multiple $-2(4,1)$?
What is the scalar multiple $-2(4,1)$?
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$(-8,-2)$. Multiply each component: $-2(4)=-8$, $-2(1)=-2$.
$(-8,-2)$. Multiply each component: $-2(4)=-8$, $-2(1)=-2$.
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What is the scalar multiple $-\frac{3}{4}(12,-8)$?
What is the scalar multiple $-\frac{3}{4}(12,-8)$?
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$(-9,6)$. Multiply: $- rac{3}{4}(12)=-9$, $- rac{3}{4}(-8)=6$.
$(-9,6)$. Multiply: $- rac{3}{4}(12)=-9$, $- rac{3}{4}(-8)=6$.
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What is the scalar $c$ if $c(-5,1)=(15,-3)$?
What is the scalar $c$ if $c(-5,1)=(15,-3)$?
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$c=-3$. Since $-3(-5)=15$ and $-3(1)=-3$, $c=-3$.
$c=-3$. Since $-3(-5)=15$ and $-3(1)=-3$, $c=-3$.
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Identify the scalar $c$ if $c(6,9)=(2,3)$.
Identify the scalar $c$ if $c(6,9)=(2,3)$.
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$c=\frac{1}{3}$. Since $ rac{1}{3}(6)=2$ and $ rac{1}{3}(9)=3$, $c= rac{1}{3}$.
$c=\frac{1}{3}$. Since $ rac{1}{3}(6)=2$ and $ rac{1}{3}(9)=3$, $c= rac{1}{3}$.
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On a coordinate plane, what does multiplying by $c$ do to each component of $(v_x,v_y)$?
On a coordinate plane, what does multiplying by $c$ do to each component of $(v_x,v_y)$?
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Multiply both components by $c$. Scalar multiplication applies to each component.
Multiply both components by $c$. Scalar multiplication applies to each component.
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On a coordinate plane, if $\vec{v}$ goes from $(0,0)$ to $(2,3)$, where does $-2\vec{v}$ end?
On a coordinate plane, if $\vec{v}$ goes from $(0,0)$ to $(2,3)$, where does $-2\vec{v}$ end?
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$(-4,-6)$. Scale and reverse: $-2(2)=-4$ and $-2(3)=-6$.
$(-4,-6)$. Scale and reverse: $-2(2)=-4$ and $-2(3)=-6$.
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State the component-wise formula for multiplying a vector $\langle v_x, v_y \rangle$ by a scalar $c$.
State the component-wise formula for multiplying a vector $\langle v_x, v_y \rangle$ by a scalar $c$.
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$c\langle v_x, v_y \rangle = \langle cv_x, cv_y \rangle$. Multiply each component by the scalar separately.
$c\langle v_x, v_y \rangle = \langle cv_x, cv_y \rangle$. Multiply each component by the scalar separately.
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What happens to a vector’s direction when it is multiplied by a positive scalar $c>0$?
What happens to a vector’s direction when it is multiplied by a positive scalar $c>0$?
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Direction stays the same. Positive scalars preserve the vector's original direction.
Direction stays the same. Positive scalars preserve the vector's original direction.
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Which option describes $\frac{1}{3}\vec{v}$ geometrically for nonzero $\vec{v}$?
Which option describes $\frac{1}{3}\vec{v}$ geometrically for nonzero $\vec{v}$?
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Same direction, one-third the magnitude. Positive fraction less than 1 shrinks but doesn't reverse.
Same direction, one-third the magnitude. Positive fraction less than 1 shrinks but doesn't reverse.
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Identify the scalar multiple: Compute $\frac{1}{2}\langle 6, -10 \rangle$ component-wise.
Identify the scalar multiple: Compute $\frac{1}{2}\langle 6, -10 \rangle$ component-wise.
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$\langle 3, -5 \rangle$. Multiply each component: $\frac{1}{2}(6)=3$ and $\frac{1}{2}(-10)=-5$.
$\langle 3, -5 \rangle$. Multiply each component: $\frac{1}{2}(6)=3$ and $\frac{1}{2}(-10)=-5$.
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Identify the scalar multiple: Compute $-\frac{3}{4}\langle 8, 12 \rangle$ component-wise.
Identify the scalar multiple: Compute $-\frac{3}{4}\langle 8, 12 \rangle$ component-wise.
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$\langle -6, -9 \rangle$. Multiply: $-\frac{3}{4}(8)=-6$ and $-\frac{3}{4}(12)=-9$.
$\langle -6, -9 \rangle$. Multiply: $-\frac{3}{4}(8)=-6$ and $-\frac{3}{4}(12)=-9$.
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What is the magnitude of $c\vec{v}$ in terms of $|c|$ and $|\vec{v}|$?
What is the magnitude of $c\vec{v}$ in terms of $|c|$ and $|\vec{v}|$?
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$|c\vec{v}|=|c|,|\vec{v}|$. Magnitude scales by the absolute value of the scalar.
$|c\vec{v}|=|c|,|\vec{v}|$. Magnitude scales by the absolute value of the scalar.
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Compute the scaled endpoint: If $\vec{v}=\langle 2, -1 \rangle$, what is $4\vec{v}$?
Compute the scaled endpoint: If $\vec{v}=\langle 2, -1 \rangle$, what is $4\vec{v}$?
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$\langle 8, -4 \rangle$. Multiply each component by 4: $4(2)=8$ and $4(-1)=-4$.
$\langle 8, -4 \rangle$. Multiply each component by 4: $4(2)=8$ and $4(-1)=-4$.
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Compute the scaled endpoint: If $\vec{v}=\langle -3, 7 \rangle$, what is $-1\vec{v}$?
Compute the scaled endpoint: If $\vec{v}=\langle -3, 7 \rangle$, what is $-1\vec{v}$?
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$\langle 3, -7 \rangle$. Multiplying by $-1$ reverses the vector: $-1(-3)=3$, $-1(7)=-7$.
$\langle 3, -7 \rangle$. Multiplying by $-1$ reverses the vector: $-1(-3)=3$, $-1(7)=-7$.
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Compute the scalar: If $\vec{v}=\langle 3, -2 \rangle$ and $c\vec{v}=\langle 12, -8 \rangle$, what is $c$?
Compute the scalar: If $\vec{v}=\langle 3, -2 \rangle$ and $c\vec{v}=\langle 12, -8 \rangle$, what is $c$?
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$c=4$. Since $12=4(3)$ and $-8=4(-2)$, the scalar is $c=4$.
$c=4$. Since $12=4(3)$ and $-8=4(-2)$, the scalar is $c=4$.
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Compute the scalar: If $\vec{v}=\langle -4, 6 \rangle$ and $c\vec{v}=\langle 2, -3 \rangle$, what is $c$?
Compute the scalar: If $\vec{v}=\langle -4, 6 \rangle$ and $c\vec{v}=\langle 2, -3 \rangle$, what is $c$?
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$c=-\frac{1}{2}$. Since $2=-\frac{1}{2}(-4)$ and $-3=-\frac{1}{2}(6)$, $c=-\frac{1}{2}$.
$c=-\frac{1}{2}$. Since $2=-\frac{1}{2}(-4)$ and $-3=-\frac{1}{2}(6)$, $c=-\frac{1}{2}$.
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Find the missing vector: If $c=3$ and $c\vec{v}=\langle 9, -12 \rangle$, what is $\vec{v}$?
Find the missing vector: If $c=3$ and $c\vec{v}=\langle 9, -12 \rangle$, what is $\vec{v}$?
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$\langle 3, -4 \rangle$. Divide each component by 3: $\vec{v}=\langle 9/3, -12/3 \rangle$.
$\langle 3, -4 \rangle$. Divide each component by 3: $\vec{v}=\langle 9/3, -12/3 \rangle$.
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Find the missing vector: If $c=-2$ and $c\vec{v}=\langle -10, 14 \rangle$, what is $\vec{v}$?
Find the missing vector: If $c=-2$ and $c\vec{v}=\langle -10, 14 \rangle$, what is $\vec{v}$?
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$\langle 5, -7 \rangle$. Divide by $-2$: $\vec{v}=\langle -10/(-2), 14/(-2) \rangle$.
$\langle 5, -7 \rangle$. Divide by $-2$: $\vec{v}=\langle -10/(-2), 14/(-2) \rangle$.
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Identify the scalar multiple: Compute $-2\langle 4, -3 \rangle$ component-wise.
Identify the scalar multiple: Compute $-2\langle 4, -3 \rangle$ component-wise.
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$\langle -8, 6 \rangle$. Multiply each component: $-2(4)=-8$ and $-2(-3)=6$.
$\langle -8, 6 \rangle$. Multiply each component: $-2(4)=-8$ and $-2(-3)=6$.
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Identify the error: A student claims $2\langle 1, -4 \rangle=\langle 1, -8 \rangle$. What is the correct result?
Identify the error: A student claims $2\langle 1, -4 \rangle=\langle 1, -8 \rangle$. What is the correct result?
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$\langle 2, -8 \rangle$. Student forgot to multiply the first component by 2.
$\langle 2, -8 \rangle$. Student forgot to multiply the first component by 2.
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What happens to a vector’s direction when it is multiplied by a negative scalar $c<0$?
What happens to a vector’s direction when it is multiplied by a negative scalar $c<0$?
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Direction reverses (points opposite). Negative scalars flip the vector to point the opposite way.
Direction reverses (points opposite). Negative scalars flip the vector to point the opposite way.
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What is the geometric effect on magnitude when multiplying a vector by a scalar $c$?
What is the geometric effect on magnitude when multiplying a vector by a scalar $c$?
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Magnitude is multiplied by $|c|$. The length scales by the absolute value of the scalar.
Magnitude is multiplied by $|c|$. The length scales by the absolute value of the scalar.
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What is the result of scalar multiplication $0\vec{v}$ for any vector $\vec{v}$?
What is the result of scalar multiplication $0\vec{v}$ for any vector $\vec{v}$?
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$\vec{0}$. Multiplying any vector by zero gives the zero vector.
$\vec{0}$. Multiplying any vector by zero gives the zero vector.
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