2.3 Algebra of Vectors

Three displacement vectors $\overset{\rightarrow}{A}$, $\overset{\rightarrow}{B}$, and $\overset{\rightarrow}{F}$ (the reference) are specified by their magnitudes A = 10.00, B = 7.00, and F = 20.00, respectively, and by their respective direction angles with the horizontal direction $\alpha = 35 °$, $\beta = −110 °$, and $\varphi = 110 °$. The physical units of the magnitudes are centimeters. Use the analytical method to find vector $\overset{\rightarrow}{G} = \overset{\rightarrow}{A} + 2 \overset{\rightarrow}{B} - \overset{\rightarrow}{F}$. Verify that
G = 28.15 cm and that $\theta_{G} = −68.65 °$.

Suppose that Balto in the example leaves the game to attend to more important matters, but Astro, Clifford, and Dug continue playing. Astro and Clifford's pull on the toy does not change, but Dug runs around and bites on the toy in a different place. With how big a force and in what direction must Dug pull on the toy now to balance out the combined pulls from Clifford and Astro? Illustrate this s…

Verify that vector $\hat{v}$ obtained in the example is indeed a unit vector by computing its magnitude. If the convoy in the example was moving across a desert flatland—that is, if the third component of its velocity was zero—what is the unit vector of its direction of motion? Which geographic direction does it represent?

For vectors $\overset{\rightarrow}{B} = − \hat{i} - 4 \hat{j}$ and $\overset{\rightarrow}{A} = −3 \hat{i} - 2 \hat{j}$, calculate (a) $\overset{\rightarrow}{A} + \overset{\rightarrow}{B}$ and its magnitude and direction angle, and (b) $\overset{\rightarrow}{A} - \overset{\rightarrow}{B}$ and its magnitude and direction angle.

Given two displacement vectors $\overset{\rightarrow}{A} = \left(3.00 \hat{i} - 4.00 \hat{j} + 4.00 \hat{k}\right) \text{m}$ and $\overset{\rightarrow}{B} = \left(2.00 \hat{i} + 3.00 \hat{j} - 7.00 \hat{k}\right) \text{m}$, find the displacements and their magnitudes for (a) $\overset{\rightarrow}{C} = \overset{\rightarrow}{A} + \overset{\rightarrow}{B}$ and (b) $\overset{\rightarrow}{D} = 2 \overset{\rightarrow}{A} - \overset{\rightarrow}{B}$.

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 she is blown along the following straight lines: 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 due 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 $10.0 °$ n…

Given the vectors in the preceding figure, find vector $\overset{\rightarrow}{R}$ that solves equations (a) $\overset{\rightarrow}{D} + \overset{\rightarrow}{R} = \overset{\rightarrow}{F}$ and (b) $\overset{\rightarrow}{C} - 2 \overset{\rightarrow}{D} + 5 \overset{\rightarrow}{R} = 3 \overset{\rightarrow}{F}$. Assume the +x-axis is horizontal to the right.

An adventurous dog strays from home, runs three blocks east, two blocks north, and one block east, one block north, and two blocks west. Assuming that each block is about a 100 yd, use the analytical method to find the dog’s net displacement vector, its magnitude, and its direction. Assume the +x-axis is to the east. How would your answer be affected if each block was about 100 m?

Given the displacement vector $\overset{\rightarrow}{D} = \left(3 \hat{i} - 4 \hat{j}\right) \text{m},$ find the displacement vector $\overset{\rightarrow}{R}$ so that $\overset{\rightarrow}{D} + \overset{\rightarrow}{R} = −4 D \hat{j}$.

At one point in space, the direction of the electric field vector is given in the Cartesian system by the unit vector $\hat{E} = 1 / \sqrt{5} \hat{i} - 2 / \sqrt{5} \hat{j}$. If the magnitude of the electric field vector is E = 400.0 V/m, what are the scalar components $E_{x}$, $E_{y}$, and $E_{z}$ of the electric field vector $\overset{\rightarrow}{E}$ at this point? What is the direction angle $\theta_{E}$ of the electric field vector at this point?

In the control tower at a regional airport, an air traffic controller monitors two aircraft that travel in straight paths directly away from the control tower. One plane is a cargo carrier Boeing 747 climbing at $10 °$ above the horizontal, and moving $30 °$ north of west. At some moment in time, the controller notes that the Boeing is at an altitude of 2500 m and the DC-3 is at an altitude of 3000 m. The…