Precalculus Flashcards: Add Vectors In Different Ways

What this deck covers

This deck focuses on Add Vectors In Different Ways, 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: What is the component-wise sum formula for $\langle a,b\rangle+\langle c,d\rangle$?

Answer: $\langle a+c,\,b+d\rangle$. Add corresponding components: first with first, second with second.

Flashcard 2: Find the magnitude $\|\langle 5,12\rangle\|$.

Answer: $13$. $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.

Flashcard 3: What is the commutative property of vector addition written with vectors $\vec{u}$ and $\vec{v}$?

Answer: $\vec{u}+\vec{v}=\vec{v}+\vec{u}$. Order doesn't matter in vector addition.

Flashcard 4: What does it mean to add vectors end-to-end (head-to-tail) in geometry?

Answer: Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.

Flashcard 5: Find $\lVert\langle 6,8\rangle\rVert$.

Answer: $10$. Use $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 6: What property states that $\vec{u}+\vec{v}$ equals $\vec{v}+\vec{u}$?

Answer: Commutative property of vector addition. Vector addition order doesn't matter, like regular addition.

Flashcard 7: What is the additive identity for vectors (the vector $\vec{0}$) in $\mathbb{R}^2$?

Answer: $\vec{0}=\langle 0,0\rangle$. The zero vector has both components equal to zero.

Flashcard 8: If $\vec{u}=\langle 2,1\rangle$ and $\vec{v}=\langle -1,5\rangle$, find $\vec{u}+\vec{v}$.

Answer: $\langle 1,\,6\rangle$. Add components: $\langle 2+(-1), 1+5\rangle = \langle 1, 6\rangle$.

Flashcard 9: If $\vec{u}=\langle 4,0\rangle$ and $\vec{v}=\langle 0,3\rangle$, find $\lVert\vec{u}+\vec{v}\rVert$.

Answer: $5$. First add to get $\langle 4,3\rangle$, then $\sqrt{4^2+3^2}=\sqrt{25}=5$.

Flashcard 10: What is the additive inverse of $\vec{v}=\langle a,b\rangle$?

Answer: $-\vec{v}=\langle -a,-b\rangle$. Negate both components to get the opposite vector.

Flashcard 11: Find the magnitude $\|\langle 6,8\rangle\|$.

Answer: $10$. $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 12: What inequality relates $\|\vec{u}+\vec{v}\|$ to $\|\vec{u}\|$ and $\|\vec{v}\|$ (triangle inequality)?

Answer: $\|\vec{u}+\vec{v}\|\le \|\vec{u}\|+\|\vec{v}\|$. The magnitude of a sum never exceeds the sum of magnitudes.

Flashcard 13: What is the associative property of vector addition for $\vec{u}$, $\vec{v}$, and $\vec{w}$?

Answer: $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$. Grouping doesn't affect the result.

Flashcard 14: What is the magnitude formula for $\vec{v}=\langle a,b\rangle$?

Answer: $\lVert\vec{v}\rVert=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to the components.

Flashcard 15: What property states that $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$?

Answer: Associative property of vector addition. Grouping doesn't affect the result when adding multiple vectors.

Flashcard 16: What inequality relates $\lVert\vec{u}+\vec{v}\rVert$ to $\lVert\vec{u}\rVert$ and $\lVert\vec{v}\rVert$?

Answer: $\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. The triangle inequality: sum magnitude ≤ magnitude sum.

Flashcard 17: Find $\langle 3,-2\rangle+\langle -5,4\rangle$.

Answer: $\langle -2,\,2\rangle$. Add components: $3+(-5)=-2$ and $-2+4=2$.

Flashcard 18: What is the component-wise subtraction formula for $\langle a,b\rangle - \langle c,d\rangle$?

Answer: $\langle a-c,\,b-d\rangle$. Subtract corresponding components: first minus first, second minus second.

Flashcard 19: Identify the correct statement about magnitudes: $\lVert\vec{u}+\vec{v}\rVert$ vs. $\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$.

Answer: Typically $\lVert\vec{u}+\vec{v}\rVert\ne\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$. Triangle inequality shows sum magnitude usually less than magnitude sum.

Flashcard 20: Find $\vec{u}+\vec{v}$ if $\vec{u}$ goes from $(0,0)$ to $(2,-1)$ and $\vec{v}$ goes from $(0,0)$ to $(-3,4)$.

Answer: $\langle -1,\,3\rangle$. $\vec{u}=\langle 2,-1\rangle$ and $\vec{v}=\langle -3,4\rangle$ sum component-wise.

Flashcard 21: What is the additive inverse of $\vec{v}=\langle a,b\rangle$?

Answer: $-\vec{v}=\langle -a,\,-b\rangle$. Negate each component to get the opposite vector.

Flashcard 22: What is the triangle (end-to-end) rule for adding $\vec{u}$ and $\vec{v}$?

Answer: Place tail of $\vec{v}$ at head of $\vec{u}$; sum is tail-to-head. Connect vectors tip-to-tail; resultant goes from start to end.

Flashcard 23: What is the additive identity vector, written as a component vector in $\mathbb{R}^2$?

Answer: $\langle 0,0\rangle$. The zero vector leaves any vector unchanged when added.

Flashcard 24: If $\vec{v}=\langle a,b\rangle$, what is $\vec{v}+(-\vec{v})$?

Answer: $\vec{0}$. A vector plus its negative always gives the zero vector.

Flashcard 25: Find $\langle 3,-2\rangle + \langle -5,4\rangle$ using component-wise addition.

Answer: $\langle -2,\,2\rangle$. $3+(-5)=-2$ and $-2+4=2$.

Flashcard 26: When does $\|\vec{u}+\vec{v}\|=\|\vec{u}\|+\|\vec{v}\|$ hold exactly?

Answer: When $\vec{u}$ and $\vec{v}$ point in the same direction. Parallel vectors with same orientation achieve equality.

Flashcard 27: What is the component-wise addition formula for $\langle a,b\rangle + \langle c,d\rangle$?

Answer: $\langle a+c,\,b+d\rangle$. Add corresponding components: first with first, second with second.

Flashcard 28: Find the vector $\overrightarrow{AB}$ for $A(1,-3)$ and $B(6,2)$.

Answer: $\langle 5,\,5\rangle$. Calculate $\langle 6-1, 2-(-3)\rangle = \langle 5, 5\rangle$.

Flashcard 29: What is the vector from point $A(x_1,y_1)$ to point $B(x_2,y_2)$ in component form?

Answer: $\langle x_2-x_1,\,y_2-y_1\rangle$. Subtract initial point coordinates from terminal point coordinates.

Flashcard 30: Compute $\lVert\langle 4,0\rangle\rVert+\lVert\langle 0,3\rangle\rVert$.

Answer: $7$. Sum individual magnitudes: $\sqrt{16}+\sqrt{9}=4+3=7$.

Flashcard 31: What is the parallelogram rule statement for $\vec{u}+\vec{v}$?

Answer: Diagonal from common tail of adjacent sides $\vec{u}$ and $\vec{v}$. Form parallelogram; sum is the diagonal from shared starting point.

Flashcard 32: Identify the correct statement about magnitudes: is $\|\vec{u}+\vec{v}\|$ typically equal to $\|\vec{u}\|+\|\vec{v}\|$?

Answer: No; typically $\|\vec{u}+\vec{v}\|\ne \|\vec{u}\|+\|\vec{v}\|$. Triangle inequality shows equality is rare.

Flashcard 33: Find $\langle -1,7\rangle - \langle 4,-3\rangle$ using component-wise subtraction.

Answer: $\langle -5,\,10\rangle$. $-1-4=-5$ and $7-(-3)=10$.

Flashcard 34: When does equality hold in $\lVert\vec{u}+\vec{v}\rVert\le\lVert\vec{u}\rVert+\lVert\vec{v}\rVert$?

Answer: When $\vec{u}$ and $\vec{v}$ point in the same direction. Equality occurs when vectors are parallel with same orientation.

Flashcard 35: What is the magnitude formula for a vector $\vec{v}=\langle a,b\rangle$?

Answer: $\|\vec{v}\|=\sqrt{a^2+b^2}$. Use the Pythagorean theorem on the components.

Flashcard 36: What is the parallelogram rule for $\vec{u}+\vec{v}$ in geometric vector addition?

Answer: Draw both from same tail; sum is diagonal of the parallelogram. Vectors form adjacent sides; sum is the diagonal.

Flashcard 37: Compute $\|\langle 1,2\rangle+\langle 2,1\rangle\|$.

Answer: $\sqrt{18}$. Sum is $\langle 3,3\rangle$; magnitude is $\sqrt{9+9}=\sqrt{18}$.