Add Vectors in Different Ways - Precalculus

1. What is the component-wise sum formula for $\langle a,b\rangle+\langle c,d\rangle$?

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

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

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

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

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

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

   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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    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.

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

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

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

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

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

    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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Study these flashcards