2.1 Scalars and Vectors

Two motorboats named Alice and Bob are moving on a lake. Given the information about their velocity vectors in each of the following situations, indicate whether their velocity vectors are equal or otherwise. (a) Alice moves north at 6 knots and Bob moves west at 6 knots. (b) Alice moves west at 6 knots and Bob moves west at 3 knots. (c) Alice moves northeast at 6 knots and Bob moves south at 3 k…

A cave diver enters a long underwater tunnel. When her displacement with respect to the entry point is 20 m, she accidentally drops her camera, but she doesn’t notice it missing until she is some 6 m farther into the tunnel. She swims back 10 m but cannot find the camera, so she decides to end the dive. How far from the entry point is she? Taking the positive direction out of the tunnel, what is…

Using the three displacement vectors $\overset{\rightarrow}{A}$, $\overset{\rightarrow}{B}$, and $\overset{\rightarrow}{F}$ in the figure, choose a convenient scale, and use a ruler and a protractor to find vector $\overset{\rightarrow}{G}$ given by the vector equation $\overset{\rightarrow}{G} = \overset{\rightarrow}{A} + 2 \overset{\rightarrow}{B} - \overset{\rightarrow}{F}$.

A weather forecast states the temperature is predicted to be $−5 ° \text{C}$ the following day. Is this temperature a vector or a scalar quantity? Explain.

Give a specific example of a vector, stating its magnitude, units, and direction.

Suppose you add two vectors $\overset{\rightarrow}{A}$ and $\overset{\rightarrow}{B}$. What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Is it possible for two vectors of different magnitudes to add to zero? Is it possible for three vectors of different magnitudes to add to zero? Explain.

When a 10,000-m runner competing on a 400-m track crosses the finish line, what is the runner’s net displacement? Can this displacement be zero? Explain.

Can a magnitude of a vector be negative?

If two vectors are equal, what can you say about their components? What can you say about their magnitudes? What can you say about their directions?

A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again…

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? Use a graphical method.

A delivery driver starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use a graphical method to find their net displacement vector.

In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km and $45.0 °$ north of west, then 4.70 km and $60.0 °$ south of east, then 1.30 km and $25.0 °$ south of west, then 5.10 km straight east, then 1.70 km and $5.00 °$ east of north, then 7.20 km and $55.0 °$ south of west, and finally 2.80 km and $10.0 °$ no…

A pedestrian walks 6.0 km east and then 13.0 km north. Use a graphical method to find the pedestrian’s resultant displacement and geographic direction.