Precalculus Flashcards: Show Scalar Multiplication Visually

What this deck covers

This deck focuses on Show Scalar Multiplication Visually, 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: Identify the scalar $c$ if $c(6,9)=(2,3)$.

Answer: $c=\frac{1}{3}$. Since rac{1}{3}(6)=2 and rac{1}{3}(9)=3, c=rac{1}{3}.

Flashcard 2: Identify the scalar multiple: Compute $-2\langle 4, -3 \rangle$ component-wise.

Answer: $\langle -8, 6 \rangle$. Multiply each component: $-2(4)=-8$ and $-2(-3)=6$.

Flashcard 3: Which option describes $\frac{1}{3}\vec{v}$ geometrically for nonzero $\vec{v}$?

Answer: Same direction, one-third the magnitude. Positive fraction less than 1 shrinks but doesn't reverse.

Flashcard 4: What happens to a vector's direction when it is multiplied by a scalar $c>0$?

Answer: Direction stays the same. Positive scalars preserve the vector's direction.

Flashcard 5: Compute the scaled endpoint: If $\vec{v}=\langle 2, -1 \rangle$, what is $4\vec{v}$?

Answer: $\langle 8, -4 \rangle$. Multiply each component by 4: $4(2)=8$ and $4(-1)=-4$.

Flashcard 6: Identify the correct statement: If $c=-5$, how does $c\vec{v}$ compare to $\vec{v}$?

Answer: Opposite direction and $5$ times as long. Negative scalar reverses direction; $|-5|=5$ scales magnitude.

Flashcard 7: What happens to a vector's direction when it is multiplied by a negative scalar $c<0$?

Answer: Direction reverses (points opposite). Negative scalars flip the vector to point the opposite way.

Flashcard 8: What is the result of scalar multiplication $0\vec{v}$ for any vector $\vec{v}$?

Answer: $\vec{0}$. Multiplying any vector by zero gives the zero vector.

Flashcard 9: What is the geometric effect on magnitude when multiplying a vector by a scalar $c$?

Answer: Magnitude is multiplied by $|c|$. The length scales by the absolute value of the scalar.

Flashcard 10: What is the scalar $c$ if $c(-5,1)=(15,-3)$?

Answer: $c=-3$. Since $-3(-5)=15$ and $-3(1)=-3$, $c=-3$.

Flashcard 11: Find the missing vector: If $c=3$ and $c\vec{v}=\langle 9, -12 \rangle$, what is $\vec{v}$?

Answer: $\langle 3, -4 \rangle$. Divide each component by 3: $\vec{v}=\langle 9/3, -12/3 \rangle$.

Flashcard 12: What is the scalar multiple $3(2,-5)$?

Answer: $(6,-15)$. Multiply each component: $3(2)=6$, $3(-5)=-15$.

Flashcard 13: What is the scalar multiple $-\frac{3}{4}(12,-8)$?

Answer: $(-9,6)$. Multiply: -rac{3}{4}(12)=-9, -rac{3}{4}(-8)=6.

Flashcard 14: What is the scalar $c$ if $c(3,-2)=(12,-8)$?

Answer: $c=4$. Since $4(3)=12$ and $4(-2)=-8$, $c=4$.

Flashcard 15: State the component-wise formula for multiplying a vector $\langle v_x, v_y \rangle$ by a scalar $c$.

Answer: $c\langle v_x, v_y \rangle = \langle cv_x, cv_y \rangle$. Multiply each component by the scalar separately.

Flashcard 16: Compute the scalar: If $\vec{v}=\langle 3, -2 \rangle$ and $c\vec{v}=\langle 12, -8 \rangle$, what is $c$?

Answer: $c=4$. Since $12=4(3)$ and $-8=4(-2)$, the scalar is $c=4$.

Flashcard 17: Identify the error: A student claims $2\langle 1, -4 \rangle=\langle 1, -8 \rangle$. What is the correct result?

Answer: $\langle 2, -8 \rangle$. Student forgot to multiply the first component by 2.

Flashcard 18: What happens to a vector's magnitude when it is multiplied by a scalar $c$?

Answer: Magnitude is multiplied by $|c|$. The scalar's absolute value scales the length.

Flashcard 19: What happens to a vector's direction when it is multiplied by a scalar $c<0$?

Answer: Direction reverses. Negative scalars flip the vector to point opposite.

Flashcard 20: What happens to a vector's direction when it is multiplied by a positive scalar $c>0$?

Answer: Direction stays the same. Positive scalars preserve the vector's original direction.

Flashcard 21: What is the magnitude of $c\vec{v}$ in terms of $|c|$ and $\|\vec{v}\|$?

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

Flashcard 22: What is the component-wise rule for scalar multiplication $c(v_x, v_y)$?

Answer: $c(v_x, v_y) = (cv_x, cv_y)$. Multiply each component by the scalar $c$.

Flashcard 23: On a coordinate plane, if $\vec{v}$ goes from $(0,0)$ to $(2,3)$, where does $-2\vec{v}$ end?

Answer: $(-4,-6)$. Scale and reverse: $-2(2)=-4$ and $-2(3)=-6$.

Flashcard 24: Identify the scalar multiple: Compute $-\frac{3}{4}\langle 8, 12 \rangle$ component-wise.

Answer: $\langle -6, -9 \rangle$. Multiply: $-\frac{3}{4}(8)=-6$ and $-\frac{3}{4}(12)=-9$.

Flashcard 25: Identify the scalar multiple: Compute $\frac{1}{2}\langle 6, -10 \rangle$ component-wise.

Answer: $\langle 3, -5 \rangle$. Multiply each component: $\frac{1}{2}(6)=3$ and $\frac{1}{2}(-10)=-5$.

Flashcard 26: Find the missing vector: If $c=-2$ and $c\vec{v}=\langle -10, 14 \rangle$, what is $\vec{v}$?

Answer: $\langle 5, -7 \rangle$. Divide by $-2$: $\vec{v}=\langle -10/(-2), 14/(-2) \rangle$.

Flashcard 27: Identify the scalar multiple: Compute $3\langle -2, 5 \rangle$ component-wise.

Answer: $\langle -6, 15 \rangle$. Multiply each component: $3(-2)=-6$ and $3(5)=15$.

Flashcard 28: What is the scalar multiple $-2(4,1)$?

Answer: $(-8,-2)$. Multiply each component: $-2(4)=-8$, $-2(1)=-2$.

Flashcard 29: On a coordinate plane, what does multiplying by $c$ do to each component of $(v_x,v_y)$?

Answer: Multiply both components by $c$. Scalar multiplication applies to each component.

Flashcard 30: Compute the scaled endpoint: If $\vec{v}=\langle -3, 7 \rangle$, what is $-1\vec{v}$?

Answer: $\langle 3, -7 \rangle$. Multiplying by $-1$ reverses the vector: $-1(-3)=3$, $-1(7)=-7$.

Flashcard 31: Compute the scalar: If $\vec{v}=\langle -4, 6 \rangle$ and $c\vec{v}=\langle 2, -3 \rangle$, what is $c$?

Answer: $c=-\frac{1}{2}$. Since $2=-\frac{1}{2}(-4)$ and $-3=-\frac{1}{2}(6)$, $c=-\frac{1}{2}$.