2.4 Products of Vectors

For the vectors given in the reference, find the scalar products $\overset{\rightarrow}{A} \cdot \overset{\rightarrow}{B}$ and $\overset{\rightarrow}{F} \cdot \overset{\rightarrow}{C}$.

Find the angle between forces $\overset{\rightarrow}{F}_{1}$ and $\overset{\rightarrow}{F}_{3}$ in the example.

How much work is done by the first dog and by the second dog in the example on the displacement in the example?

For the vectors given in the reference, find the vector products $\overset{\rightarrow}{A} \times \overset{\rightarrow}{B}$ and $\overset{\rightarrow}{C} \times \overset{\rightarrow}{F}$.

Given two vectors $\overset{\rightarrow}{A} = − \hat{i} + \hat{j}$ and $\overset{\rightarrow}{B} = 3 \hat{i} - \hat{j}$, find (a) $\overset{\rightarrow}{A} \times \overset{\rightarrow}{B}$, (b) $\left|\overset{\rightarrow}{A} \times \overset{\rightarrow}{B}\right|$, (c) the angle between $\overset{\rightarrow}{A}$ and $\overset{\rightarrow}{B}$, and (d) the angle between $\overset{\rightarrow}{A} \times \overset{\rightarrow}{B}$ and vector $\overset{\rightarrow}{C} = \hat{i} + \hat{k}$.

What is wrong with the following expressions? How can you correct them? (a) $C = \overset{\rightarrow}{A} \overset{\rightarrow}{B}$, (b) $\overset{\rightarrow}{C} = \overset{\rightarrow}{A} \overset{\rightarrow}{B}$, (c) $C = \overset{\rightarrow}{A} \times \overset{\rightarrow}{B}$, (d) $C = A \overset{\rightarrow}{B}$, (e) $C + 2 \overset{\rightarrow}{A} = B$, (f) $\overset{\rightarrow}{C} = A \times \overset{\rightarrow}{B}$, (g) $\overset{\rightarrow}{A} \cdot \overset{\rightarrow}{B} = \overset{\rightarrow}{A} \times \overset{\rightarrow}{B}$, (h) $\overset{\rightarrow}{C} = 2 \overset{\rightarrow}{A} \cdot \overset{\rightarrow}{B}$, (i) $C = \overset{\rightarrow}{A} / \overset{\rightarrow}{B}$, and (j) $C = \overset{\rightarrow}{A} / B$.

If the dot product of two vectors vanishes, what can you say about their directions?

Assuming the +x-axis is horizontal to the right for the vectors in the preceding figure, find (a) the component of vector $\overset{\rightarrow}{A}$ along vector $\overset{\rightarrow}{C}$, (b) the component of vector $\overset{\rightarrow}{C}$ along vector $\overset{\rightarrow}{A}$, (c) the component of vector $\hat{i}$ along vector $\overset{\rightarrow}{F}$, and (d) the component of vector $\overset{\rightarrow}{F}$ along vector $\hat{i}$.

Find the angles that vector $\overset{\rightarrow}{D} = \left(2.0 \hat{i} - 4.0 \hat{j} + \hat{k}\right) \text{m}$ makes with the x-, y-, and z- axes.

Assuming the +x-axis is horizontal to the right for the vectors in the previous figure, find the following vector products: (a) $\overset{\rightarrow}{A} \times \overset{\rightarrow}{C}$, (b) $\overset{\rightarrow}{A} \times \overset{\rightarrow}{F}$, (c) $\overset{\rightarrow}{D} \times \overset{\rightarrow}{C}$, (d) $\overset{\rightarrow}{A} \times \left(\overset{\rightarrow}{F} + 2 \overset{\rightarrow}{C}\right)$, (e) $\hat{i} \times \overset{\rightarrow}{B}$, (f) $\hat{j} \times \overset{\rightarrow}{B}$, (g) $\left(3 \hat{i} - \hat{j}\right) \times \overset{\rightarrow}{B}$, and (h) $\hat{B} \times \overset{\rightarrow}{B}$.

For the vectors in the earlier figure, find (a) $\left(\overset{\rightarrow}{A} \times \overset{\rightarrow}{F}\right) \cdot \overset{\rightarrow}{D}$, (b) $\left(\overset{\rightarrow}{A} \times \overset{\rightarrow}{F}\right) \cdot \left(\overset{\rightarrow}{D} \times \overset{\rightarrow}{B}\right)$, and (c) $\left(\overset{\rightarrow}{A} \cdot \overset{\rightarrow}{F}\right) \left(\overset{\rightarrow}{D} \times \overset{\rightarrow}{B}\right)$.

You fly $32.0 \text{km}$ in a straight line in still air in the direction $35.0 °$ south of west. (a) Find the distances you would have to fly due south and then due west to arrive at the same point. (b) Find the distances you would have to fly first in a direction $45.0 °$ south of west and then in a direction $45.0 °$ west of north. Note these are the components of the displacement along a different set of axes—namely, the one rot…

If the polar coordinates of a point are $\left(r , \varphi\right)$ and its rectangular coordinates are $\left(x , y\right)$, determine the polar coordinates of the following points: (a) (−x, y), (b) (−2x, −2y), and (c) (3x, −3y).

Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4.76 nautical miles (nmi) in a direction $37 °$ north of east to Noi. From Noi, it sails $69 °$ west of north to Poi. On its return leg from Poi, it sails $28 °$ east of south. What distance does the boat sail between Noi and Poi? What distance does it sai…

Show that when $\overset{\rightarrow}{A} + \overset{\rightarrow}{B} = \overset{\rightarrow}{C}$, then $C^{2} = A^{2} + B^{2} + 2 A B \text{cos} \varphi$, where $\varphi$ is the angle between vectors $\overset{\rightarrow}{A}$ and $\overset{\rightarrow}{B}$.

A skater glides along a circular path of radius 5.00 m in clockwise direction. When he coasts around one-half of the circle, starting from the west point, find (a) the magnitude of his displacement vector and (b) how far he actually skated. (c) What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the west point?

If the velocity vector of a polar bear is $\overset{\rightarrow}{u} = \left(−18.0 \hat{i} - 13.0 \hat{j}\right) \text{km} / \text{h}$, how fast and in what geographic direction is it heading? Here, $\hat{i}$ and $\hat{j}$ are directions to geographic east and north, respectively.

A diver explores a shallow reef off the coast of Belize. She initially swims 90.0 m north, makes a turn to the east and continues for 200.0 m, then follows a big grouper for 80.0 m in the direction $30 °$ north of east. In the meantime, a local current displaces her by 150.0 m south. Assuming the current is no longer present, in what direction and how far should she now swim to come back to the point w…

Vectors $\overset{\rightarrow}{A}$ and $\overset{\rightarrow}{B}$ are two orthogonal vectors in the xy-plane and they have identical magnitudes. If $\overset{\rightarrow}{A} = 3.0 \hat{i} + 4.0 \hat{j}$, find $\overset{\rightarrow}{B}$.

What is the component of the force vector $\overset{\rightarrow}{G} = \left(3.0 \hat{i} + 4.0 \hat{j} + 10.0 \hat{k}\right) \text{N}$ along the force vector $\overset{\rightarrow}{H} = \left(1.0 \hat{i} + 4.0 \hat{j}\right) \text{N}$?