Vectors - AP Precalculus

1. What is the definition of a vector?

   A vector is a quantity with both magnitude and direction. This distinguishes vectors from scalars which only have magnitude.

2. What is meant by the term 'orthogonal vectors'?

   Vectors with a dot product of zero. Perpendicular vectors meet at right angles.

3. Find the projection of \begin{bmatrix} 3 \ 4 \bmatrix} onto \begin{bmatrix} 6 \ 8 \bmatrix}.

   \begin{bmatrix} 3 \ 4 \bmatrix}. Both vectors are parallel, so projection equals the first vector.

4. Find the sum of vectors $\begin{bmatrix} -2 \\ 5 \end{bmatrix}$ and $\begin{bmatrix} 7 \\ -3 \end{bmatrix}$.

   $\begin{bmatrix} 5 \\ 2 \end{bmatrix}$. Add components: $(-2+7, 5-3) = (5,2)$.

5. Find the vector sum of $\textbf{a} = \begin{bmatrix} 3 \\ -2 \end{bmatrix}$ and $\textbf{b} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}$.

   $\begin{bmatrix} 2 \\ 2 \end{bmatrix}$. Add corresponding components: $(3-1, -2+4) = (2,2)$.

6. What operation is used to find a vector's component along another vector?

   Projection. Projects one vector onto the direction of another.

7. State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.

   $\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.

8. State the formula for the dot product of \textbf{u} = \begin{bmatrix} a \ b \bmatrix} and \textbf{v} = \begin{bmatrix} c \ d \bmatrix}.

   $a \times c + b \times d$. Multiply corresponding components and sum the results.

9. Find the angle between $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

   $90^\circ$. Standard unit vectors are perpendicular to each other.

10. Find the sum of vectors \begin{bmatrix} -2 \ 5 \bmatrix} and \begin{bmatrix} 7 \ -3 \bmatrix}.

    \begin{bmatrix} 5 \ 2 \bmatrix}. Add components: $(-2+7, 5-3) = (5,2)$.

11. Find the scalar projection of \begin{bmatrix} 3 \ 4 \bmatrix} on \begin{bmatrix} 4 \ 3 \bmatrix}.

    $\frac{24}{5}$. Scalar projection: $\frac{\textbf{u} \cdot \textbf{v}}{|\textbf{v}|} = \frac{24}{5}$.

12. What does a negative scalar multiplication do to a vector?

    Reverses the direction of the vector. Changes direction but preserves magnitude.

13. Find the magnitude of \begin{bmatrix} 7 \ 24 \bmatrix}.

    $25$. Use $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25$.

14. What is the direction of a vector if its components are all equal?

    The direction is along the line $y=x=z$. Equal components create a vector along the main diagonal.

15. Find the angle between \begin{bmatrix} 1 \ 0 \bmatrix} and \begin{bmatrix} 0 \ 1 \bmatrix}.

    $90^\text{°}$. Standard unit vectors are perpendicular to each other.

16. Find the magnitude of vector \begin{bmatrix} 3 \ 4 \bmatrix}.

    $5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

17. What is the magnitude of the zero vector?

    Zero. The zero vector has no length by definition.

18. State the associative property of vector addition.

    ($\textbf{u} + \textbf{v}) + \textbf{w} = \textbf{u} + (\textbf{v} + \textbf{w})$. Grouping doesn't affect vector addition results.

19. State the formula for the dot product of $\textbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \end{bmatrix}$.

    $a \times c + b \times d$. Multiply corresponding components and sum the results.

20. Find the dot product of \begin{bmatrix} 1 \ 3 \bmatrix} and \begin{bmatrix} 4 \ -2 \bmatrix}.

    $-2$. Calculate: $(1)(4) + (3)(-2) = 4 - 6 = -2$.

21. How do you find the angle $\theta$ between two vectors $\textbf{u}$ and $\textbf{v}$?

    $\theta = \text{acos}\bigg(\frac{\textbf{u} \bullet \textbf{v}}{|\textbf{u}||\textbf{v}|}\bigg)$. Uses the dot product formula and inverse cosine function.

22. Find the magnitude of vector $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$

    $5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$

23. Identify the property: $\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$.

    Commutative property of vector addition. Vector addition is commutative like regular addition.

24. What is the result of scalar multiplication of a vector by zero?

    A zero vector. Multiplying by zero eliminates all magnitude and direction.

25. What is the magnitude of \begin{bmatrix} 0 \ 0 \ 0 \bmatrix}?

    Zero. The zero vector has zero magnitude by definition.

26. Determine if vectors \begin{bmatrix} 2 \ 3 \bmatrix} and \begin{bmatrix} -4 \ -6 \bmatrix} are parallel.

    Yes, they are parallel. The second vector is $-2$ times the first vector.

27. Find the unit vector of \textbf{v} = \begin{bmatrix} 5 \ 12 \bmatrix}.

    \begin{bmatrix} \frac{5}{13} \ \frac{12}{13} \bmatrix}. Divide by magnitude $13$ to get unit vector.

28. What is the inverse of vector $\textbf{v}$?

    The vector $-\textbf{v}$. (Negation). The additive inverse of a vector.

29. What does it mean if the dot product of two vectors is zero?

    The vectors are orthogonal (perpendicular). Zero dot product indicates 90° angle between vectors.

30. Find the projection of $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ onto $\begin{bmatrix} 6 \\ 8 \end{bmatrix}$.

    $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$. Both vectors are parallel, so projection equals the first vector.

31. What is the result of multiplying a vector by a scalar $k$?

    A vector in the same direction if $k > 0$, opposite if $k < 0$. Scales magnitude and may reverse direction based on sign.

32. What is a linear combination of vectors?

    A sum of scalar multiples of vectors. Combines vectors with scalar coefficients.

33. What does it mean for vectors to be linearly dependent?

    One vector is a linear combination of the others. One vector can be expressed using the others.

34. Find the vector sum of \textbf{a} = \begin{bmatrix} 3 \ -2 \bmatrix} and \textbf{b} = \begin{bmatrix} -1 \ 4 \bmatrix}.

    \begin{bmatrix} 2 \ 2 \bmatrix}. Add corresponding components: $(3-1, -2+4) = (2,2)$.

35. What is the magnitude of $\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$?

    Zero. The zero vector has zero magnitude by definition.

36. Find the dot product of $\begin{bmatrix} 1 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ -2 \end{bmatrix}$.

    $-2$. Calculate: $(1)(4) + (3)(-2) = 4 - 6 = -2$.

37. Which condition indicates that two vectors are parallel?

    One vector is a scalar multiple of the other. Parallel vectors point in the same or opposite directions.

38. What is the result of adding a vector to its negative?

    The zero vector. A vector plus its additive inverse equals zero vector.

39. How is a vector typically represented in a plane?

    As an arrow from an initial point to a terminal point. The arrow shows both direction and magnitude visually.

40. Determine if vectors $\begin{bmatrix} 2 \\ 3 \\ \end{bmatrix}$ and $\begin{bmatrix} -4 \\ -6 \\ \end{bmatrix}$ are parallel.

    Yes, they are parallel. The second vector is $-2$ times the first vector.

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