Solving Problems with Vectors and Velocity - Precalculus

1. With $\vec{r}_0=\langle 1,2\rangle$ and $\vec{v}=\langle 3,-1\rangle$, what is $\vec{r}(4)$?

   $\langle 13,-2\rangle$. $\langle 1,2\rangle+4\langle 3,-1\rangle=\langle 1+12,2-4\rangle$.

2. What is the formula for the projection of $\vec{v}$ onto $\vec{u}$ (vector projection)?

   $\text{proj}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{\|\vec{u}\|^2}\,\vec{u}$. Projects $\vec{v}$ onto $\vec{u}$ using dot product and scaling.

3. What is the unit vector in the direction of $\vec{v}$ (assuming $\vec{v}\neq\vec{0}$)?

   $\hat{v}=\frac{\vec{v}}{\|\vec{v}\|}$. Divide vector by its magnitude to get length 1.

4. Find $\operatorname{proj}_{\langle 1,0\rangle}\langle 3,4\rangle$.

   $\langle 3,0\rangle$. Projects onto x-axis: keeps x-component, zeros y.

5. State the formula for the vector projection of $\vec{a}$ onto a nonzero vector $\vec{b}$.

   $\operatorname{proj}_{\vec{b}}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|^2}\vec{b}$. Scales $\vec{b}$ by the scalar projection ratio.

6. What is the formula for the scalar component of $\vec{v}$ in the direction of $\vec{u}$?

   $\text{comp}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{\|\vec{u}\|}$. Gives the signed length of the projection of $\vec{v}$ onto $\vec{u}$.

7. Find position at $t=3$ if $\vec{r}_0=\langle 2,-1\rangle$ and $\vec{v}=\langle -4,5\rangle$.

   $\langle -10,\,14\rangle$. $\vec{r}(3) = \langle 2,-1\rangle + 3\langle -4,5\rangle = \langle 2-12, -1+15\rangle$.

8. What is the distance traveled in time $t$ at constant velocity vector $\vec{v}$ (assume $t\ge 0$)?

   $\text{distance}=\|\vec{v}\|t$. Speed times time equals distance traveled.

9. What is the component form of the displacement vector from $(x_1,y_1)$ to $(x_2,y_2)$?

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

10. State the constant-velocity position formula for a particle with $\vec{r}(0)=\vec{r}_0$ and velocity $\vec{v}$.

    $\vec{r}(t)=\vec{r}_0+t\vec{v}$. Linear motion: start at $\vec{r}_0$, move by $t\vec{v}$.

11. State the formula for the scalar projection of $\vec{a}$ onto a nonzero vector $\vec{b}$.

    $\operatorname{comp}_{\vec{b}}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|}$. Measures signed length of $\vec{a}$'s shadow on $\vec{b}$.

12. What condition on $\vec{u}\cdot\vec{v}$ shows two nonzero vectors are perpendicular?

    $\vec{u}\cdot\vec{v}=0$. Perpendicular vectors have a dot product of zero.

13. Identify the condition for two vectors $\vec{a}$ and $\vec{b}$ to be perpendicular using a dot product.

    $\vec{a}\cdot\vec{b}=0$. Perpendicular vectors have dot product zero.

14. State the formula for scalar multiplication: $k\langle a,b\rangle$.

    $\langle ka,\,kb\rangle$. Multiply each component by the scalar $k$.

15. Find a unit vector in the direction of $\vec{v}=\langle 3,4\rangle$.

    $\left\langle \frac{3}{5},\,\frac{4}{5}\right\rangle$. $\|\vec{v}\| = 5$, so $\hat{v} = \frac{1}{5}\langle 3,4\rangle$.

16. Find the speed of $\vec{v}=\langle 6,8\rangle$.

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

17. State the formula for the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$.

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

18. What is the position vector formula for constant velocity: initial $\vec{r}_0$ and velocity $\vec{v}$?

    $\vec{r}(t)=\vec{r}_0+t\vec{v}$. Position equals initial position plus displacement over time.

19. What is the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$?

    $\|\vec{v}\|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to find the length of the vector.

20. Find the speed (magnitude) of the velocity vector $\langle 5,12\rangle$.

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

21. Compute the dot product $\langle 2,-1\rangle\cdot\langle 3,4\rangle$.

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

22. What is the formula for the angle $\theta$ between nonzero vectors using the dot product?

    $\cos\theta=\frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}$. Rearranged from $\vec{a}\cdot\vec{b}=\|\vec{a}\|\|\vec{b}\|\cos\theta$.

23. State the dot product formula for $\vec{a}=\langle a_1,a_2\rangle$ and $\vec{b}=\langle b_1,b_2\rangle$.

    $\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2$. Multiply corresponding components and add.

24. Find the magnitude of $\vec{v}=\langle 3,4\rangle$.

    $5$. $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

25. State the formula for vector addition in components: $\langle a,b\rangle+\langle c,d\rangle$.

    $\langle a+c,\,b+d\rangle$. Add corresponding components to get the resultant vector.

26. Identify whether $\langle 1,2\rangle$ and $\langle 4,-2\rangle$ are perpendicular.

    $\text{Yes, since }\langle 1,2\rangle\cdot\langle 4,-2\rangle=0$. $(1)(4) + (2)(-2) = 4 - 4 = 0$, so they are perpendicular.

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

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

28. What is the unit vector in the direction of a nonzero vector $\vec{v}$?

    $\hat{v}=\frac{\vec{v}}{\|\vec{v}\|}$. Divide the vector by its magnitude to get a vector of length 1.

29. Compute the dot product $\langle 2,-1\rangle\cdot\langle 5,4\rangle$.

    $6$. $(2)(5) + (-1)(4) = 10 - 4 = 6$.

30. What is the dot product formula for $\vec{u}=\langle a,b\rangle$ and $\vec{v}=\langle c,d\rangle$?

    $\vec{u}\cdot\vec{v}=ac+bd$. Multiply corresponding components and add the products.

31. What is the displacement vector after time $t$ with constant velocity $\vec{v}$?

    $\Delta\vec{r}=t\vec{v}$. Displacement equals velocity times time for constant motion.

32. Find the resultant of perpendicular velocities $\langle 5,0\rangle$ and $\langle 0,12\rangle$.

    $\langle 5,12\rangle$. Add components: $\langle 5+0,0+12\rangle=\langle 5,12\rangle$.

33. Identify the speed if velocity is $\vec{v}=\langle v_x,v_y\rangle$.

    $\text{speed}=\|\vec{v}\|=\sqrt{v_x^2+v_y^2}$. Speed is the magnitude of the velocity vector.

34. Find $\text{proj}_{\vec{u}}\vec{v}$ for $\vec{u}=\langle 1,0\rangle$ and $\vec{v}=\langle 3,4\rangle$.

    $\langle 3,\,0\rangle$. $\frac{\vec{v}\cdot\vec{u}}{\|\vec{u}\|^2}\vec{u} = \frac{3}{1}\langle 1,0\rangle = \langle 3,0\rangle$.

35. Find the resultant velocity: $\langle 3, -2\rangle+\langle -5, 7\rangle$.

    $\langle -2,\,5\rangle$. Add components: $(3-5, -2+7) = \langle -2, 5\rangle$.

36. What is the angle relation for dot product using magnitudes and angle $\theta$?

    $\vec{u}\cdot\vec{v}=\|\vec{u}\|\,\|\vec{v}\|\cos\theta$. Relates dot product to magnitudes and the angle between vectors.

37. A boat has velocity $\langle 4,0\rangle$ in still water; current is $\langle 0,3\rangle$. Find ground velocity.

    $\langle 4,3\rangle$. Add boat and current vectors component-wise.

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