## Section 3.9 Vectors Bootcamp

### Subsection 3.9.1 Representing Vectors

###### Problem 3.9.1. Magnitude and Direction of a Vector in 2D from Components.

Follow the link: Checkpoint 3.1.5.

###### Problem 3.9.2. Components of a Vector from Magnitude and Direction in 2D.

Follow the link: Checkpoint 3.1.6.

###### Problem 3.9.3. Components of Velocity Vector of a Cricket Ball in a Horizontal-Vertical Plane.

Follow the link: Checkpoint 3.1.7.

###### Problem 3.9.4. Components of Velocity Vector of a Soccer Ball in a Horizontal-Vertical Plane.

Follow the link: Checkpoint 3.1.8.

###### Problem 3.9.5. Magnitude and Direction of Velocity from Components.

Follow the link: Checkpoint 3.1.9.

###### Problem 3.9.6. Magnitude and Direction of Velocity from Components - 2.

Follow the link: Checkpoint 3.1.10.

###### Problem 3.9.7. Magnitude and Direction of Force.

Follow the link: Checkpoint 3.1.12.

###### Problem 3.9.8. Practice Computing Magnitudes and Stating Directions.

Follow the link: Checkpoint 3.1.25.

###### Problem 3.9.9. Components of a Force Vector from Magnitude and Direction in First Quadrant.

Follow the link: Checkpoint 3.1.23.

###### Problem 3.9.10. Direction in the Second Quadrant.

Follow the link: Checkpoint 3.1.14.

###### Problem 3.9.11. Direction in the Third Quadrant.

Follow the link: Checkpoint 3.1.17.

###### Problem 3.9.12. Direction in the Fourth Quadrant.

Follow the link: Checkpoint 3.1.20.

###### Problem 3.9.13. Components of a Force Vector from Magnitude and Direction in the Fourth Quadrant.

Follow the link: Checkpoint 3.1.27.

###### Problem 3.9.14. Magnitude and Direction of a Force Vector from Components in the Second Quadrant.

Follow the link: Checkpoint 3.1.29.

###### Problem 3.9.15. Unit Vector in an an Arbitrary Direction.

Follow the link: Checkpoint 3.2.7.

### Subsection 3.9.2 Adding Vectors

###### Problem 3.9.16. Adding Forces Using an Organizing Table.

Follow the link: Example 3.3.10.

###### Problem 3.9.17. Adding Two Forces Given in Component Form.

Follow the link: Checkpoint 3.3.7.

###### Problem 3.9.18. Adding Two Forces Given in Magnitude-Direction Form.

Follow the link: Checkpoint 3.3.8.

###### Problem 3.9.19. Adding Two Force Vectors.

Follow the link: Checkpoint 3.3.9.

###### Problem 3.9.20. Adding Three Velocity Vectors.

Follow the link: Checkpoint 3.3.13.

###### Problem 3.9.21. Adding Vectors Graphically.

Follow the link: Checkpoint 3.3.3.

###### Problem 3.9.22. Negative of a Vector.

Follow the link: Checkpoint 3.4.1.

###### Problem 3.9.23. Finding the Third Force that will Balance Two Other Forces Given in Component Form.

Follow the link: Checkpoint 3.4.2.

###### Problem 3.9.24. Finding the Third Force that will Balance Two Other Forces Given in Magnitude-Direction Form.

Follow the link: Checkpoint 3.4.4.

### Subsection 3.9.3 Multiplying Vectors

###### Problem 3.9.25. Finding Angle Between Two Vectors.

Follow the link: Checkpoint 3.6.2.

###### Problem 3.9.26. Vectors Whose Dot Pruduct is Zero.

Follow the link: Checkpoint 3.6.3.

###### Problem 3.9.27. More Practice on Dot Product.

Follow the link: Checkpoint 3.6.4.

###### Problem 3.9.28. Practice with the Right Hand Rule.

Follow the link: Checkpoint 3.7.2.

###### Problem 3.9.29. Magnetic Force on a Current Carrying Wire.

Follow the link: Checkpoint 3.7.3.

###### Problem 3.9.30. Cross Product Using Components of Vectors.

Follow the link: Checkpoint 3.7.4.

###### Problem 3.9.31. Computing Dot and Cross Products of Displacement and Force Vectors.

Follow the link: Checkpoint 3.7.5.

###### Problem 3.9.32. Computing Dot and Cross Products of Displacement and Force Vectors.

Follow the link: Checkpoint 3.7.6.

###### Problem 3.9.33. Dot and Cross Products of Perpendicularly Directed Vectors.

Follow the link: Checkpoint 3.7.8.

###### Problem 3.9.34. Unit Vector in the Direction of a Vector.

Follow the link: Checkpoint 3.2.9.

### Subsection 3.9.4 Miscellaneous

###### Problem 3.9.35. Prove the Law of Cosine.

In a triangle \(\triangle ABC \text{,}\) let \(a,\, b,\) and \(c \) denote the lengths of the sides, and C denote the angle between sides \(a\) and \(b\text{.}\) Use vector concepts to prove the following relation.

Use a triangle of vectors oriented such that \(\vec c = \vec a + \vec b \text{.}\)

In the problem statement.

Consider a triangle of vectors along the sides of the triangle as shown in Figure 3.9.36. From this figure, we get the following vector relation.

Now, taking the dot product of each side with itself, we get

We can rewrite the last term in terms of the angle between vectors \(\vec a\) and \(\vec b \text{,}\) which is just \(180^{\circ} -C\text{.}\) Therefore,

Using this in Eq. (3.9.1) gives the result we seek.

###### Problem 3.9.37. Prove the Law of Sines.

In a triangle \(\triangle ABC \text{,}\) let \(a,\, b,\) and \(c \) denote the lengths of the sides, and A, B, C denote the angles opposite to the corresponding sides as shown in the figure. Use vector concepts to prove the following relations.

Examine height of the triangle.

In the problem statement.

Look at the construction in the figure below, where height from the tip with angle \(\langle A \) to the the (extended) base on the other side is shown.

Since

equating the two expressions for the height gives

Therefore,

Similar construction from the tip at angle \(\langle C \) will show

The two equations taken together prove the statement of the law of sines.

###### Problem 3.9.38. Relative Velocity.

Suppose you and your friend are on a large ship. Suppose the ship has a velocity of \(3 \) m/s pointed towards the positive \(x \) axis as seen from the shore. Suppose your friend is moving with respect to you (i.e., the ship) at a velocity of \(4 \) m/s towards the positive \(y \) axis. What is your friend's velocity vector with respect to the shore. Find both component, and the magnitude and direction forms.

Note that a good notation will help. For instance, we can label each velocity with two subscripts, one for the object and the second the object with respect to which we have that velocity. Thus, let \(\vec v_{SG} \) will be velocity vector of the ship with respect to the ground. This is given in componen form, \(\vec v_{SG} = (3 \text{ m/s}, 0)\) in the \(xy\)-plane. Similarly, \(\vec v_{FS} = (0, 4 \text{ m/s})\text{.}\) And, we want \(\vec v_{FG} \text{.}\)

Use \(\vec v_{FG} = \vec v_{FS} + \vec v_{SG}\text{.}\)

\(5 \text{ m/s at } 53.1^{\circ} \text{ counterclockwise from } +x \text{ axis}.\)

The relation between velocities, as seen in the figure provided, can be formally written as

We can use the vector addition of components to obtain the components of \(\vec v_{FG} \text{.}\)

From this we can find the magnitude and angle \(\theta \text{.}\)

Since the vector \((3 \text{ m/s}, 4 \text{ m/s}) \) is in the first qudrant, the angle we foound is the counterclockwise angle from the \(+x\) axis.

###### Problem 3.9.39. Crossing a River.

In order for you to cross a river exactly on the opposite side from one shore, you have to point the boat slightly up stream. This happens because your velocity with respect to shore is different from your velocity with respect to the river.

Let \(\vec v_{BS} \) be the velocity of the boat (B) with respect to the shore (S), \(\vec v_{BR} \) be the velocity of the boat (B) with respect to the river (R), and \(\vec v_{RS} \) be the velocity of the river stream (R) with respect to the shore (S). These vectors form a triangle as shown in the figure below. Notice that vectors \(\vec v_{BR} \) and \(\vec v_{RS} \) must add up to the vector \(\vec v_{BS} \text{.}\)

Now, the question. In order to cross the river directly on the other side, suppose you need to row the boat at speed \(5\text{ m/s} \) at \(\theta=30^{\circ}\) to directly across direction, which is the velocity of the boat with respect to the river. (a) What is the speed of the river? (b) If the river is \(100\text{ m}\) wide, how long would it take to cross the rivew?

Express velocities in coordinate form and use Eq. (3.9.2).

(a) \(2.5\text{ m/s}\text{,}\) (b) \(23.1\text{ s}\text{.}\)

(a) Let's use simpler notation in our calculations. Let \(v \) denote the speed of the river with respect to the shore. Let \(u\) be the speed of the boat with respect to the shore. Let \(x\) axis be in the direction of the river flow and \(y\) axis directly across. Then, Eq. (3.9.2), written using components form of the vectors is (suppressing the units):

This gives us two equations, one along the \(x \) axis and the other along the \(y \) axis.

This gives \(u = 4.33\text{ m/s} \) and \(v = 2.5 \text{ m/s}\text{.}\) We will use \(u \) in part (b). The river is flowing at speed \(v = 2.5\text{ m/s} \text{.}\)

(b) The boat moves at speed \(u = 4.33 \) m/s across the river which is \(d = 100\text{ m} \text{.}\) Therefore, it will take

###### Problem 3.9.40. Sum and Difference of a Time-dependent Vector and a Constant Vector.

Two vectors \(\vec A\) and \(\vec B\) are defined as \(\vec A=a\hat i\) and \(\vec B = a\left(\cos\,\omega t\,\hat i + \sin\,\omega t\,\hat j \right)\text{,}\) where \(a\) is a constant and \(\omega = \pi/6\text{ rad s}^{-1}\text{.}\)

If \(\left| \vec A + \vec B\right| = \sqrt{3} \left| \vec A - \vec B\right|\) at time \(t=\tau\) for the first time, find then value of \(\tau\) in seconds. (Adpated from Indian JEE Advanced, 2018)

Use definition of square of amplitude of a vector.

\(2.0\text{ sec}\text{.}\)

This question can be answered by performing the algebra on the given condition.

Therefore, square of \(\left| \vec A + \vec B\right| = \sqrt{3} \left| \vec A - \vec B\right|\) at \(t=\tau\) becomes

We can cancel out \(a^2\) from every term, expand \((1+\cos x)^2\) and use \(\sin^2 x + \cos^2 x = 1\text{,}\) where \(x=\omega \tau\text{.}\)

This yields

Since \(\cos^{-1}\left(\frac{1}{2} \right) = \frac{\pi}{3}\text{ rad}\text{,}\) we have

###### Problem 3.9.41. Decomposing a Vector into Components Parallel and Perpendicular to a Unit Vector.

Often, we are interested in how much of a vector is along some given unit vector and along some direction perpendicular to that unit vector. Let \(\hat u\) be a given unit vector. Show the following relation holds for an arbitrary vector \(\vec A\text{.}\)

where the parallel and perpendicular components of \(\vec A\) are

###### Problem 3.9.42. Practice with a Friend.

Prove the following identities for two arbitrary vectors \(\vec A\) and \(\vec B\text{.}\)

No solution provided.

###### Problem 3.9.43. Practice with a Friend.

Prove the following identitiy for three arbitrary vectors \(\vec A\text{,}\) \(\vec B\text{,}\) and \(\vec C\text{,}\) called the cab-bac equation.

Maybe you want to show that \(x\) component of each side comes out the same.

No solution provided.

###### Problem 3.9.44. Practice with a Friend.

A parallelopiped can be described by stating three vectors along its edges as shown in Figure 3.9.45. Just as area of the base here could be obtained from the magnitude of vector product \(|\vec B \times \vec C|\text{,}\) the volume of the parallelopiped can be obtained by a scalar triple prroduct.

Prove that volume is indeed given by this formula.

Also show that you get the same answer by cyclically rotating the vector names, e.g.,

No solution provided.

###### Problem 3.9.46. Practice with a Friend: Prove a plane geometry result using vectors..

If the diagonals of a parallelogram are equal, then the sides must be perpendicular to each other.