8. Vectors

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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)

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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)

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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)

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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)

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

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

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

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

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

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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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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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°

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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°

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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°

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

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

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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

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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)

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

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

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

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

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

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

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

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

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

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

〈15, -8〉

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

〈-4, 4√3〉

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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)

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c + d

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

θ = 50°, |v| = 26

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

θ = 27° 30' |v| = 15.4

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

|u| = 12, |v| = 20, θ = 27°

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

|u| = 20, |v| = 30, θ = 30°

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

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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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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°

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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′

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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.

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.

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

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

v - u

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

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

-5v

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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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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?

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

- 2u + 4v

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

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〉

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

〈2, 0〉

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

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Find the dot product for each pair of vectors.

4i, 5i - 9j

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

〈5, 7〉

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Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

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Given vectors u and v, find: v - 3u.

u = 2i, v = i + j

Given vectors u and v, find: 2u.

u = 〈-1, 2〉, v = 〈3, 0〉

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Given vectors u and v, find: 2u + 3v.

u = 〈-1, 2〉, v = 〈3, 0〉

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Given vectors u and v, find: v - 3u.

u = 〈-1, 2〉, v = 〈3, 0〉

A force of 30.0 lb is required to hold an 80.0-lb pressure washer on an incline. What angle does the incline make with the horizontal?

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Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.

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A crate is supported by two ropes. One rope makes an angle of 46° 20′ with the horizontal and has a tension of 89.6 lb on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.

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A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

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Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.

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Starting at point X, a ship sails 15.5 km on a bearing of 200°, then turns and sails 2.4 km on a bearing of 320°. Find the distance of the ship from point X.

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Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.

(3u) • v

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Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.

u • v - u • w

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Given vectors u and v, find: 2u.

u = 2i, v = i + j

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

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

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

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A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?

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One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

Solve each problem. See Examples 5 and 6.

Bearing and Ground Speed of a Plane An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from a direction of 114.0°. Find the resulting bearing and ground speed of the plane.

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A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane.

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A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57° 40′. A wind is blowing from the south at 27.1 mph. Find the bearing she should fly, and find the plane's ground speed.

A plane is headed due south with an airspeed of 192 mph. A wind from a direction of 78.0° is blowing at 23.0 mph. Find the ground speed and resulting bearing of the plane.

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a + b