8. Vectors

Determine if the following statement is true or false: Temperature is a vector.

123

Determine if the following statement is true or false: Acceleration is a vector.

127

Determine if the following statement is true or false: The vectors $v$ and $-v$ point in the same direction.

129

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $u ⃗+v ⃗$

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $u ⃗-v ⃗$ .

144

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $\frac12u⃗+v⃗$ .

106

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $-2u ⃗+3v ⃗$ .

117

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.

230

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

203

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

207

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.

198

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

211

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

220

If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

214

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = 3i + j

204

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w

183

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j

221

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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165

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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170

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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187

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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172

Solve each problem. See Examples 5 and 6.

Distance and Direction of a Motorboat A motorboat sets out in the direction N 80° 00′ E. The speed of the boat in still water is 20.0 mph. If the current is flowing directly south, and the actual direction of the motorboat is due east, find the speed of the current and the actual speed of the motorboat.

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185

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = -6i - 2j

206

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = -4i

186

In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-4, -4), P₂ = (6, 2)

226

In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-8, 6), P₂ = (-2, 3)

277

In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-1, 7), P₂ = (-7, -7)

224

In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-3, 4), P₂ = (6, 4)

249

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. u + v

196

In Exercises 22–24, sketch each vector as a position vector and find its magnitude. v = -3i - 4j

210

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. u - v

183

In Exercises 22–24, sketch each vector as a position vector and find its magnitude. v = -3j

202

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. v - u

189

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (2, -1), P₂ = (5, -3)

199

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-3, 0), P₂ = (-2, -2)

199

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 5v

183

In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar. w - v

207

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. -4w

212

In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar. ||-2v||

212

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3w + 2v

174

In Exercises 31–32, find the unit vector that has the same direction as the vector v. v = -i + 2j

220

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3v - 4w

200

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||2u||

214

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||w - u||

225

In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 6i

233

In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 4j

232

In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 2j

483

In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = i + j

311

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°

327

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 12, θ = 225°

221

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°

242

In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. 4u - (2v - w)

197

In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. ||u + v||² - ||u - v||²

253

In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j

191

In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = (4i - 2j) - (4i - 8j)

199

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

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185

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

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155

Use the figure to find each vector: - u. Use vector notation as in Example 4.

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176

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

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179

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

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175

Use the figure to find each vector: - u. Use vector notation as in Example 4.

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176

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

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167

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

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164

Use the figure to find each vector: - u. Use vector notation as in Example 4.

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184

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈15, -8〉

191

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈-4, 4√3〉

167

Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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a + (b + c)

193

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c + d

191

Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.

θ = 50°, |v| = 26

268

Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.

θ = 27° 30' |v| = 15.4

171

For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 12, |v| = 20, θ = 27°

188

For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 20, |v| = 30, θ = 30°

218

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth.

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188

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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153

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth.

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444

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 250 and 450 newtons, forming an angle of 85°

237

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 116 and 139 lb, forming an angle of 140° 50′

222

Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.

210

Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.

215

Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

207

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

v - u

206

Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

228

A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angle of 41° 10′ with the first force. Find the magnitudes of the second force and of the resultant.

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252

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

-5v

181

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.

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194

A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?

226

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

- 2u + 4v

164

Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.

210

To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?

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Write each vector in the form a i + b j.

〈6, -3〉

224

Write each vector in the form a i + b j.

〈2, 0〉

194

CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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-b

153

Find the dot product for each pair of vectors.

4i, 5i - 9j

173