2.3 Algebra of vectors  (Page 6/8)

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

A particle undergoes three consecutive displacements given by vectors ${\overrightarrow{D}}_1=(3.0\widehat{i}-4.0\widehat{j}-2.0\widehat{k})\text{mm}$ , ${\overrightarrow{D}}_2=(1.0\widehat{i}-7.0\widehat{j}+4.0\widehat{k})\text{mm}$ , and ${\overrightarrow{D}}_3=(\mathrm{-7.0}\widehat{i}+4.0\widehat{j}+1.0\widehat{k})\text{mm}$ . (a) Find the resultant displacement vector of the particle. (b) What is the magnitude of the resultant displacement? (c) If all displacements were along one line, how far would the particle travel?

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

A small plane flies $40.0\text{km}$ in a direction $60\text{°}$ north of east and then flies $30.0\text{km}$ in a direction $15\text{°}$ north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its 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 she is blown along the following straight lines: 2.50 km and $45.0\text{°}$ north of west, then 4.70 km and $60.0\text{°}$ south of east, then 1.30 km and $25.0\text{°}$ south of west, then 5.10 km due east, then 1.70 km and $5.00\text{°}$ east of north, then 7.20 km and $55.0\text{°}$ south of west, and finally 2.80 km and $10.0\text{°}$ north of east. Use the analytical method to find the resultant vector of all her displacement vectors. What is its magnitude and direction?

Assuming the + x -axis is horizontal to the right for the vectors given in the following figure, use the analytical method to find the following resultants: (a) $\overrightarrow{A}+\overrightarrow{B},$ (b) $\overrightarrow{C}+\overrightarrow{B}$ , (c) $\overrightarrow{D}+\overrightarrow{F}$ , (d) $\overrightarrow{A}-\overrightarrow{B}$ , (e) $\overrightarrow{D}-\overrightarrow{F}$ , (f) $\overrightarrow{A}+2\overrightarrow{F}$ , (g) $\overrightarrow{C}-2\overrightarrow{D}+3\overrightarrow{F}$ , and (h) $\overrightarrow{A}-4\overrightarrow{D}+2\overrightarrow{F}$ .

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

A delivery man 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 the analytical method to determine the following: (a) Find his net displacement vector. (b) How far is the restaurant from the post office? (c) If he returns directly from the restaurant to the post office, what is his displacement vector on the return trip? (d) What is his compass heading on the return trip? Assume the + x -axis is to the east.

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?

If $\overrightarrow{D}=(6.00\widehat{i}-8.00\widehat{j})\text{m}$ , $\overrightarrow{B}=(\mathrm{-8.00}\widehat{i}+3.00\widehat{j})\text{m}$ , and $\overrightarrow{A}=(26.0\widehat{i}+19.0\widehat{j})\text{m}$ , find the unknown constants a and b such that $a\overrightarrow{D}+b\overrightarrow{B}+\overrightarrow{A}=\overrightarrow{0}$ .

Given the displacement vector $\overrightarrow{D}=(3\widehat{i}-4\widehat{j})\text{m,}$ find the displacement vector $\overrightarrow{R}$ so that $\overrightarrow{D}+\overrightarrow{R}=\mathrm{-4}D\widehat{j}$ .

Find the unit vector of direction for the following vector quantities: (a) Force $\overrightarrow{F}=(3.0\widehat{i}-2.0\widehat{j})\text{N}$ , (b) displacement $\overrightarrow{D}=(\mathrm{-3.0}\widehat{i}-4.0\widehat{j})\text{m}$ , and (c) velocity $\overrightarrow{v}=(\mathrm{-5.00}\widehat{i}+4.00\widehat{j})\text{m/s}$ .

At one point in space, the direction of the electric field vector is given in the Cartesian system by the unit vector $\widehat{E}=1\text{/}\sqrt{5}\widehat{i}-2\text{/}\sqrt{5}\widehat{j}$ . 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 $\overrightarrow{E}$ at this point? What is the direction angle ${\theta }_E$ of the electric field vector at this point?

A barge is pulled by the two tugboats shown in the following figure. One tugboat pulls on the barge with a force of magnitude 4000 units of force at $15\text{°}$ above the line AB (see the figure and the other tugboat pulls on the barge with a force of magnitude 5000 units of force at $12\text{°}$ below the line AB. Resolve the pulling forces to their scalar components and find the components of the resultant force pulling on the barge. What is the magnitude of the resultant pull? What is its direction relative to the line AB?

In the control tower at a regional airport, an air traffic controller monitors two aircraft as their positions change with respect to the control tower. One plane is a cargo carrier Boeing 747 and the other plane is a Douglas DC-3. The Boeing is at an altitude of 2500 m, climbing at $10\text{°}$ above the horizontal, and moving $30\text{°}$ north of west. The DC-3 is at an altitude of 3000 m, climbing at $5\text{°}$ above the horizontal, and cruising directly west. (a) Find the position vectors of the planes relative to the control tower. (b) What is the distance between the planes at the moment the air traffic controller makes a note about their positions?