Vector and Matrix Quantities: Scalar Multiplication of Vectors (CCSS.N-VM.5)
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Common Core High School Number And Quantity › Vector and Matrix Quantities: Scalar Multiplication of Vectors (CCSS.N-VM.5)
What is $3\mathbf{u}$ if $\mathbf{u}=\langle 2,-1\rangle$?
$\langle 5,-1\rangle$
$\langle 6,-3\rangle$
$\langle 6,-1\rangle$
$\langle 2,-3\rangle$
Explanation
Multiply each component by 3: $3\langle 2,-1\rangle=\langle 6,-3\rangle$. Scalar multiplication scales both components.
What is $-2\langle 3,4\rangle$?
$\langle 1,2\rangle$
$\langle -6,4\rangle$
$\langle 6,8\rangle$
$\langle -6,-8\rangle$
Explanation
Multiply both components by $-2$: $-2\langle 3,4\rangle=\langle -6,-8\rangle$. The negative reverses direction.
If $\mathbf{v}=\langle -5,2\rangle$ and $k=\tfrac{1}{2}$, what is $k\mathbf{v}$?
$\langle -\tfrac{5}{2},1\rangle$
$\langle -\tfrac{5}{2},2\rangle$
$\langle -\tfrac{3}{2},1\rangle$
$\langle -\tfrac{3}{2},\tfrac{1}{2}\rangle$
Explanation
Scale each component by $\tfrac{1}{2}$: $\tfrac{1}{2}\langle -5,2\rangle=\langle -\tfrac{5}{2},1\rangle$. This shrinks the magnitude by a factor of 2.
Which statement about scalar multiplication is true for a scalar $c$ and nonzero vector $\mathbf{v}$?
$\lVert c\mathbf{v}\rVert=c^2\lVert\mathbf{v}\rVert$
Multiplying by $-1$ does not change direction
$\lVert c\mathbf{v}\rVert=|c|\lVert\mathbf{v}\rVert$, and the direction of $c\mathbf{v}$ is along $\mathbf{v}$ if $c>0$ and opposite if $c<0$
Scalar multiplication adds $c$ to each component
Explanation
Scalar multiplication scales magnitude by $|c|$ and keeps direction for $c>0$, reverses it for $c<0$. So $\lVert c\mathbf{v}\rVert=|c|\lVert\mathbf{v}\rVert$.
Given $\mathbf{u}=\langle 7,-3\rangle$, which vector equals $-3\mathbf{u}$?
$\langle 4,-6\rangle$
$\langle -21,9\rangle$
$\langle -21,-9\rangle$
$\langle 7,-9\rangle$
Explanation
Multiply both components by $-3$: $-3\langle 7,-3\rangle=\langle -21,9\rangle$. Both components are scaled; the negative flips direction.
What is $3\mathbf{u}$ if $\mathbf{u}=\langle 2,-1\rangle$?
$\langle 5,2\rangle$
$\langle 6,-3\rangle$
$\langle 6,-1\rangle$
$\langle 6,-2\rangle$
Explanation
Multiply both components by 3: $3\langle 2,-1\rangle=\langle 6,-3\rangle$. Scalar multiplication scales each component and thus the magnitude by the same factor.
What is $-2\langle 3,4\rangle$?
$\langle 1,2\rangle$
$\langle -6,4\rangle$
$\langle -2,-8\rangle$
$\langle -6,-8\rangle$
Explanation
Multiply each component by $-2$: $-2\langle 3,4\rangle=\langle -6,-8\rangle$. The magnitude is doubled and the direction is reversed because the scalar is negative.
If $|\mathbf{v}|=5$, what is $|-3\mathbf{v}|$?
15
-15
2
5
Explanation
Use $|c\mathbf{v}|=|c|,|\mathbf{v}|$. Here $|{-3}|\cdot 5=15$.
For nonzero $\mathbf{v}$, which statement is true about the directions of $\tfrac{1}{2}\mathbf{v}$ and $-\mathbf{v}$?
Both $\tfrac{1}{2}\mathbf{v}$ and $-\mathbf{v}$ point opposite to $\mathbf{v}$.
$\tfrac{1}{2}\mathbf{v}$ reverses direction; $-\mathbf{v}$ keeps the same direction.
$\tfrac{1}{2}\mathbf{v}$ keeps the same direction; $-\mathbf{v}$ reverses direction.
Both keep the same direction as $\mathbf{v}$.
Explanation
Positive scalars keep direction and scale magnitude; negative scalars reverse direction. Thus $\tfrac{1}{2}\mathbf{v}$ points along $\mathbf{v}$, while $-\mathbf{v}$ points against $\mathbf{v}$.