1.
A hawk is flying at a speed of 20 mph in the direction [latex]65°[/latex] west of north.
Chapter 9: Vectors
Problems: #4, 6, 8, 29, 22, 24, 26, 28, 30, 32
For Problems 1–4, sketch an arrow to represent the vector. Find its components in the directions north and east.
A hawk is flying at a speed of 20 mph in the direction [latex]65°[/latex] west of north.
The island is located 36 miles from port on a bearing of [latex]160°\text{.}[/latex]
The tractor pulls with a force of 1200 pounds in the direction [latex]20°[/latex] west of south.
The current runs southeast at a speed of 4 mph.
For Problems 5–6, find the magnitude and direction of the vector.
[latex]A_x = -6,~ A_y = -9[/latex]
[latex]w_x = 15.2,~ w_y = -8.6[/latex]
For Problems 7–8, find the coordinate form of the vector.
[latex]\|{\bf{v}}\| = 2,~ \theta = 300°[/latex]
[latex]\|{\bf{v}}\| = 10,~ \theta = 225°[/latex]
For Problems 9–12,
The displacement vector from [latex](-8,-4)[/latex] to [latex](7,-1)[/latex]
The displacement vector from [latex](5,35)[/latex] to [latex](-10,15)[/latex]
This morning we began hiking from our camp 4 miles east and 2 miles south of the lodge, and this evening we are 6 miles east and 8 miles south of the lodge.
The tunnel should start 100 meters east and 400 meters north of the survey point and should end 500 meters west and 150 meters north of the survey point.
For Problems 13–16,
A fire crew is located 2 kilometers due west of the fire station. The station reports a new hot spot 6 kilometers away in the direction [latex]50°[/latex] east of north. How far is the hot spot from the fire crew, and in what direction?
A helicopter has just delivered a patient to the hospital located 15 miles northwest of the heliport. The pilot gets a call to pick up a passenger located 18 miles from the heliport on a bearing of [latex]200°\text{.}[/latex] How far is the passenger from the helicopter, and in what direction?
Red Rock is located at [latex]4.2{\bf{i}}+2.8{\bf{j}}[/latex] from the town of Dry Gulch, measured in miles, and Skull Point is located at [latex]-3.5{\bf{i}}+6.3{\bf{j}}[/latex] from Dry Gulch. How far is it from Red Rock to Skull Point, and in what direction?
A coast guard cutter is located 7 miles south and 5 miles west of port when it gets a distress call from a sailboat that reports its location as 1 mile north and 5 miles east of port. How far is it from the cutter to the sailboat, and in what direction?
For Problems 17–18,
For Problems 19–22, find the vector, where
[latex]{\bf{u}} = 4{\bf{i}}+2{\bf{j}},~~~{\bf{v}} = -3{\bf{i}}-{\bf{j}},~~~{\bf{w}} = 2{\bf{i}}-3{\bf{j}}[/latex]
[latex]{\bf{u}}-3{\bf{v}}[/latex]
[latex]{\bf{v}}-2({\bf{u}}-{\bf{w}})[/latex]
[latex]3({\bf{v}}+{\bf{w}})-{\bf{u}}[/latex]
[latex]2{\bf{u}} - 3{\bf{w}} - {\bf{v}}[/latex]
For Problems 23–26, find the vector described.
The unit vector in the same direction as [latex]2{\bf{i}}+3{\bf{j}}\text{.}[/latex]
The unit vector in the same direction as [latex]5{\bf{i}}+12{\bf{j}}\text{.}[/latex]
The vector of length 3 in the same direction as [latex]-2{\bf{i}}-5{\bf{j}}\text{.}[/latex]
The vector of magnitude 6 in the same direction as [latex]-3{\bf{i}}+2{\bf{j}}[/latex]
For Problems 27–28, find the component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}\text{.}[/latex]
[latex]{\bf{v}} = -6{\bf{i}}-{\bf{j}},~~ {\bf{w}} = 4{\bf{i}}-3{\bf{j}}[/latex]
[latex]{\bf{v}} = -2{\bf{i}}+{\bf{j}},~~ {\bf{w}} = {\bf{i}}-2{\bf{j}}[/latex]
For Problems 29–30, compute the dot product [latex]{\bf{u}} \cdot {\bf{v}}\text{.}[/latex]
[latex]{\bf{u}}= 3.8{\bf{i}}+4.8{\bf{j}},[/latex] [latex]~ {\bf{v}} = -9.2{\bf{i}}+5.6{\bf{j}}[/latex]
[latex]{\bf{u}}= -27{\bf{i}}+35{\bf{j}},[/latex] [latex]~ {\bf{v}} = -16{\bf{i}}-24{\bf{j}}[/latex]
For Problems 31–32, find the angle between the vectors.
[latex]{\bf{v}} = -4{\bf{i}}-3{\bf{j}},~~ {\bf{w}} = 4{\bf{i}}-3{\bf{j}}[/latex]
[latex]{\bf{v}} = 8{\bf{i}}-2{\bf{j}},[/latex] [latex]~ {\bf{w}} = -5{\bf{i}}-{\bf{j}}[/latex]