8. Vectors

Geometric Vectors

8. Vectors

# Geometric Vectors - Video Tutorials & Practice Problems

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concept

## Introduction to Vectors

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Problem

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

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True

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False

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Cannot be determined

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Problem

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

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True

B

False

C

Cannot be Determined

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Problem

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

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True

B

False

C

Cannot be determined

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concept

## Adding Vectors Geometrically

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Problem

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

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Problem

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

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example

## Example 1

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concept

## Multiplying Vectors By Scalars

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10

Problem

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

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Problem

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

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example

## Multiplying Vectors By Scalars Example 1

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PRACTICE PROBLEMS AND ACTIVITIES (117)

- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.
- In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.
- In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.
- In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
- If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.
- 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
- In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j
- Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropria...
- Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropria...
- Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropria...
- Find the angle between the vectors. Round to the nearest tenth of a degree. If the vectors are orthogonal, say...
- Solve each problem. See Examples 5 and 6.Distance and Direction of a Motorboat A motorboat sets out in the dir...
- In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = -6i - 2j
- In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = -4i
- In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and ...
- In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and ...
- In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and ...
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. u +...
- 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 -...
- In Exercises 22–24, sketch each vector as a position vector and find its magnitude. v = -3j
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. v -...
- In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and ...
- In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and ...
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 5v
- In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar. w - v
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. -4w
- In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar. ||-2v||
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3w ...
- In Exercises 31–32, find the unit vector that has the same direction as the vector v. v = -i + 2j
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3v ...
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||2...
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||w...
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 6i
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 4j
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 2j
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = i + j
- In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are giv...
- In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are giv...
- In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are giv...
- 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 -...
- In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the near...
- In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the near...
- Use the figure to find each vector: u + v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u - v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: - u. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u + v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u - v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: - u. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u + v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u - v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: - u. Use vector notation as in Example 4.<IMAGE>
- Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessar...
- CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sk...
- Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessar...
- Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represen...
- Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represen...
- Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.θ = 50°, |v|...
- Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.θ = 27° 30' ...
- For each pair of vectors u and v with angle θ between them, sketch the resultant.|u| = 12, |v| = 20, θ = 27°
- 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 figur...
- Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figur...
- Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the r...
- Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the r...
- Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between ...
- Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. Th...
- Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle b...
- Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.v - u
- Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forc...
- A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angl...
- Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.-5v
- A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an ang...
- A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizo...
- Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.- 2u + 4v
- Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizonta...
- To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building...
- Write each vector in the form a i + b j.〈6, -3〉
- Write each vector in the form a i + b j.〈2, 0〉
- Find the dot product for each pair of vectors.〈6, -1〉, 〈2, 5〉
- CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sk...
- Find the dot product for each pair of vectors.〈5, 2〉, 〈-4, -10〉
- Find the dot product for each pair of vectors.4i, 5i - 9j
- Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessar...
- Given vectors u and v, find: 2u + 3v.u = 2i, v = i + j
- 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〉
- Given vectors u and v, find: 2u + 3v. u = 〈-1, 2〉, v = 〈3, 0〉
- 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 m...
- Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. T...
- A crate is supported by two ropes. One rope makes an angle of 46° 20′ with the horizontal and has a tension of...
- 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....
- 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 m...
- 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 3...
- 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 32...
- Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.(3u) • v
- Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.u • v - u • w
- Given vectors u and v, find: 2u.u = 2i, v = i + j
- CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sk...
- Use the figure to find each vector: u + v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: u - v. Use vector notation as in Example 4.<IMAGE>
- Use the figure to find each vector: - v. Use vector notation as in Example 4.<IMAGE>
- 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...
- One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the fir...
- Solve each problem. See Examples 5 and 6.Bearing and Ground Speed of a Plane An airline route from San Francis...
- A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the pla...
- 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 ...
- 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 m...
- Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represen...