Vector and Matrix Quantities: Finding Components of Vectors (CCSS.N-VM.2)
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Common Core High School Number And Quantity › Vector and Matrix Quantities: Finding Components of Vectors (CCSS.N-VM.2)
Find the component form of the vector from $A(-2,5)$ to $B(4,-1)$.
$\langle 6,-6\rangle$
$\langle -6,6\rangle$
$\langle 2,-3\rangle$
$\langle -2,3\rangle$
Explanation
Subtract initial from terminal: $\langle 4-(-2),,-1-5\rangle=\langle 6,-6\rangle$.
What condition makes two vectors equal?
They have the same magnitude only.
They have the same magnitude and direction (equivalently, the same components).
They point in the same direction, regardless of length.
They have the same $x$-components only.
Explanation
Two vectors are equal iff corresponding components match: $\langle a,b\rangle=\langle c,d\rangle$ only when $a=c$ and $b=d$, which is the same as having equal magnitude and direction.
Find the component form of the vector from $P(3,-7)$ to $Q(-5,2)$.
$\langle 8,-9\rangle$
$\langle -8,-9\rangle$
$\langle -8,9\rangle$
$\langle -2,9\rangle$
Explanation
Compute $Q-P$: $\langle -5-3,,2-(-7)\rangle=\langle -8,9\rangle$.
Given $u=\langle 3,-4\rangle$, $v=\langle -3,4\rangle$, $w=\langle 3,-4\rangle$, and $t=\langle 0,5\rangle$, which vectors are equal?
$u$ and $v$
$v$ and $w$
$v$ and $t$
$u$ and $w$
Explanation
Vectors are equal if their components match; here $u$ and $w$ are both $\langle 3,-4\rangle$.
Find the component form of the vector from $S(1,-4,2)$ to $T(-3,5,-1)$.
$\langle -4,9,-3\rangle$
$\langle 4,-9,3\rangle$
$\langle -4,1,-3\rangle$
$\langle -2,9,-1\rangle$
Explanation
Subtract initial from terminal component-wise: $\langle -3-1,,5-(-4),,-1-2\rangle=\langle -4,9,-3\rangle$.
Find the component form of the vector from $P(-3,5)$ to $Q(4,-2)$.
$\langle -7,7\rangle$
$\langle 7,7\rangle$
$\langle 7,-7\rangle$
$\langle -7,-7\rangle$
Explanation
Use $\langle x_2-x_1,,y_2-y_1\rangle$. Here $\langle 4-(-3),,-2-5\rangle=\langle 7,-7\rangle$.
Given $u=\langle 2,-5\rangle$, $v=\langle -2,5\rangle$, $w=\langle 2,-5\rangle$, and $t=\langle 5,-2\rangle$, which vectors are equal?
$u$ and $w$
$u$ and $v$
$v$ and $t$
$u$ and $t$
Explanation
Vectors are equal if they have the same components (equivalently, the same magnitude and direction). Only $u$ and $w$ have identical components.
Find the component form of the vector from $C(7,-1)$ to $D(-2,9)$.
$\langle 9,10\rangle$
$\langle -9,-10\rangle$
$\langle -11,8\rangle$
$\langle -9,10\rangle$
Explanation
Compute $\langle -2-7,,9-(-1)\rangle=\langle -9,10\rangle$.
Find the component form of the vector from $E(-4,-6)$ to $F(3,-2)$.
$\langle -7,-4\rangle$
$\langle 7,-4\rangle$
$\langle 7,4\rangle$
$\langle -7,4\rangle$
Explanation
Use $\langle x_2-x_1,,y_2-y_1\rangle$: $\langle 3-(-4),,-2-(-6)\rangle=\langle 7,4\rangle$.