Magnitude and Direction of Scaled Vectors - Precalculus

1. Find $\|3\vec{v}\|$ if $\|\vec{v}\|=\frac{5}{2}$.

   $\|3\vec{v}\|=\frac{15}{2}$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=3\cdot\frac{5}{2}$.

2. What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c<0$ and $\vec{v}\neq\vec{0}$?

   Against $\vec{v}$ (opposite direction). Negative scalars reverse the vector's direction.

3. Find $\|\frac{1}{2}\mathbf{v}\|$ if $\|\mathbf{v}\|=18$.

   $9$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $\frac{1}{2} \cdot 18 = 9$.

4. If $\|c\mathbf{v}\|=24$ and $\|\mathbf{v}\|=6$, what is $|c|$?

   $|c|=4$. From $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $24 = |c| \cdot 6$, so $|c| = 4$.

5. Find $\|(-4)\vec{v}\|$ if $\|\vec{v}\|=7$.

   $\|(-4)\vec{v}\|=28$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=|-4|\cdot 7=4\cdot 7$.

6. What is the direction of $-7\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?

   Opposite direction to $\mathbf{v}$. Since $-7 < 0$, the vector points oppositely.

7. Find $\|\frac{-1}{2}\vec{v}\|$ if $\|\vec{v}\|=12$.

   $\|\frac{-1}{2}\vec{v}\|=6$. Use $\|c\vec{v}\|=|c|\|\vec{v}\|=\frac{1}{2}\cdot 12$.

8. What is the direction of $c\vec{v}$ relative to $\vec{v}$ when $c>0$ and $\vec{v}\neq\vec{0}$?

   Along $\vec{v}$ (same direction). Positive scalars preserve the vector's direction.

9. Choose the correct statement: if $0<|c|<1$, then $\|c\vec{v}\|$ is what relative to $\|\vec{v}\|$?

   Smaller: $\|c\vec{v}\|<\|\vec{v}\|$ for $\vec{v}\neq\vec{0}$. Since $0<|c|<1$, we have $|c|\|\vec{v}\|<\|\vec{v}\|$.

10. What is the direction of $0.2\mathbf{v}$ relative to $\mathbf{v}$ if $\mathbf{v}\neq\mathbf{0}$?

    Same direction as $\mathbf{v}$. Since $0.2 > 0$, the vector maintains its direction.

11. Identify the scalar $c$ that keeps $c\mathbf{v}$ pointing in the same direction as $\mathbf{v}$ (assuming $\mathbf{v}\neq\mathbf{0}$).

    Any $c>0$. Positive scalars maintain the original direction.

12. Find $\|-4\mathbf{v}\|$ if $\|\mathbf{v}\|=2.5$.

    $10$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|-4| \cdot 2.5 = 10$.

13. State the formula for the magnitude of a scalar multiple $c\vec{v}$ in terms of $c$ and $\|\vec{v}\|$.

    $\|c\vec{v}\|=|c|\,\|\vec{v}\|$. The magnitude scales by the absolute value of the scalar.

14. State the magnitude rule for a scalar multiple $c\mathbf{v}$ in terms of $\|\mathbf{v}\|$.

    $\|c\mathbf{v}\|=|c|\,\|\mathbf{v}\|$. The magnitude of a scalar multiple equals the absolute value of the scalar times the original magnitude.

15. Find $\|3\mathbf{v}\|$ if $\|\mathbf{v}\|=5$.

    $15$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|3| \cdot 5 = 15$.

16. Find $\|c\vec{v}\|$ if $c=-0.2$ and $\|\vec{v}\|=30$.

    $\|c\vec{v}\|=6$. Calculate $\|c\vec{v}\|=|-0.2|\cdot 30=0.2\cdot 30$.

17. What is the direction of $c\mathbf{v}$ relative to $\mathbf{v}$ when $c>0$ and $\mathbf{v}\neq \mathbf{0}$?

    Along $\mathbf{v}$ (same direction). Positive scalars preserve the vector's direction.

18. Compute $\|(-2)\mathbf{v}\|$ in terms of $\|\mathbf{v}\|$.

    $\|(-2)\mathbf{v}\|=2\|\mathbf{v}\|$. Since $|-2| = 2$, the magnitude doubles.

19. What is $\|c\mathbf{v}\|$ when $c=0$ (for any vector $\mathbf{v}$)?

    $\|0\mathbf{v}\|=0$. Zero scalar always produces zero magnitude regardless of the vector.

20. What is $\|\frac{1}{3}\vec{v}\|$ in terms of $\|\vec{v}\|$?

    $\|\frac{1}{3}\vec{v}\|=\frac{1}{3}\|\vec{v}\|$. Apply $\|c\vec{v}\|=|c|\|\vec{v}\|$ with $|\frac{1}{3}|=\frac{1}{3}$.

21. Find $\|c\mathbf{v}\|$ if $c=1.6$ and $\|\mathbf{v}\|=10$.

    $16$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|1.6| \cdot 10 = 1.6 \cdot 10 = 16$.

22. Find $|c|$ if $\|\vec{v}\|=8$ and $\|c\vec{v}\|=2$.

    $|c|=\frac{1}{4}$. From $\|c\vec{v}\|=|c|\|\vec{v}\|$: $2=|c|\cdot 8$.

23. Identify the direction of $(-5)\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.

    Opposite direction to $\vec{v}$. Since $-5<0$, the direction reverses.

24. What is the direction of $c\mathbf{v}$ relative to $\mathbf{v}$ when $c<0$ and $\mathbf{v}\neq \mathbf{0}$?

    Against $\mathbf{v}$ (opposite direction). Negative scalars reverse the vector's direction.

25. Find $\|-\frac{3}{5}\mathbf{v}\|$ if $\|\mathbf{v}\|=20$.

    $12$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $\frac{3}{5} \cdot 20 = 12$.

26. Find $c$ if $\|\vec{v}\|=9$, $c>0$, and $\|c\vec{v}\|=27$.

    $c=3$. Solve $27=|c|\cdot 9$ with $c>0$, so $|c|=c=3$.

27. Identify when $c\vec{v}$ has the same direction as $\vec{v}$ (assume $\vec{v}\neq\vec{0}$).

    When $c>0$. Positive scalars maintain the original vector direction.

28. Identify the scalar $c$ that reverses the direction of $\mathbf{v}$ in $c\mathbf{v}$ (assuming $\mathbf{v}\neq\mathbf{0}$).

    Any $c<0$. Negative scalars flip the vector to point oppositely.

29. What is the magnitude scale factor from $\vec{v}$ to $c\vec{v}$ (assume $\vec{v}\neq\vec{0}$)?

    Multiply by $|c|$. The formula $\|c\vec{v}\|=|c|\|\vec{v}\|$ shows magnitude scales by $|c|$.

30. Identify the direction of $\frac{7}{3}\vec{v}$ relative to $\vec{v}$, assuming $\vec{v}\neq\vec{0}$.

    Same direction as $\vec{v}$. Since $\frac{7}{3}>0$, the direction is preserved.

31. If $\|c\mathbf{v}\|=3$ and $\|\mathbf{v}\|=12$, what is $|c|$?

    $|c|=\frac{1}{4}$. From $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $3 = |c| \cdot 12$, so $|c| = \frac{1}{4}$.

32. Identify the direction of $(-1)\vec{v}$ relative to $\vec{v}$ for $\vec{v}\neq\vec{0}$.

    Opposite direction to $\vec{v}$. Multiplying by $-1$ reverses direction while preserving magnitude.

33. What is $\|2\vec{v}\|$ in terms of $\|\vec{v}\|$?

    $\|2\vec{v}\|=2\|\vec{v}\|$. Apply $\|c\vec{v}\|=|c|\|\vec{v}\|$ with $|2|=2$.

34. What is $\|c\vec{v}\|$ when $c=0$ (for any vector $\vec{v}$)?

    $\|0\vec{v}\|=0$. Zero scalar always produces zero vector with zero magnitude.

35. Find $\|c\mathbf{v}\|$ if $c=-0.3$ and $\|\mathbf{v}\|=50$.

    $15$. Apply $\|c\mathbf{v}\|=|c|\|\mathbf{v}\|$: $|-0.3| \cdot 50 = 0.3 \cdot 50 = 15$.

36. Compute $\|(-1)\mathbf{v}\|$ in terms of $\|\mathbf{v}\|$.

    $\|(-1)\mathbf{v}\|=\|\mathbf{v}\|$. Since $|-1| = 1$, the magnitude remains unchanged.

37. Identify the magnitude of $(-1)\vec{v}$ in terms of $\|\vec{v}\|$.

    $\|(-1)\vec{v}\|=\|\vec{v}\|$. Since $|-1|=1$, the magnitude remains unchanged.

38. If $\mathbf{v}\neq\mathbf{0}$ and $c\mathbf{v}$ points opposite $\mathbf{v}$, which sign must $c$ have?

    $c<0$. Vectors point oppositely only when the scalar is negative.

39. If $\|c\mathbf{v}\|=0$ and $\|\mathbf{v}\|\neq 0$, what must $c$ equal?

    $c=0$. Only $c=0$ makes $\|c\mathbf{v}\|=0$ when $\mathbf{v} \neq \mathbf{0}$.

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