Ch. 4 - Laws of Sines and Cosines; Vectors
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- In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 95, c = 125, A = 49°
Problem 30
- In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar. ||-2v||
Problem 30
- In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 3i, w = -4j
Problem 31
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3w + 2v
Problem 31
- In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 1.4, b = 2.9, A = 142°
Problem 32
- In Exercises 31–32, find the unit vector that has the same direction as the vector v. v = -i + 2j
Problem 32
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3v - 4w
Problem 33
- In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = 3i - 2j, w = i - j
Problem 33
- In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. A = 22°, b = 20 feet, c = 50 feet
Problem 34
- In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Problem 35
- In Exercises 35–38, find the exact value of the following under the given conditions: β e. cos ------- 2 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Problem 35
- In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = i + 3j, w = -2i + 5j
Problem 35
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||2u||
Problem 35
- In Exercises 35–36, the three given points are the vertices of a triangle. Solve each triangle, rounding lengths of sides to the nearest tenth and angle measures to the nearest degree. A(0, 0), B(-3, 4), C(3, -1)
Problem 35
- In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. B = 125°, a = 8 yards, c = 5 yards
Problem 36
- If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).
Problem 36
- In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = i + 2j, w = 3i + 6j
Problem 37
- In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. ||w - u||
Problem 37
- In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2
Problem 38
- In Exercises 35–38, find the exact value of the following under the given conditions: β e. cos ------- 2 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2
Problem 38
- In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. C = 102°, a = 16 meters, b = 20 meters
Problem 38
- In Exercises 37–39, find the dot product v ⋅ w. Then find the angle between v and w to the nearest tenth of a degree. v = 2i + 4j, w = 6i - 11j
Problem 38
- In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector. 5u ⋅ (3v - 4w)
Problem 39
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 6i
Problem 39
- In Exercises 40–41, use the dot product to determine whether v and w are orthogonal. v = 12i - 8j, w = 2i + 3j
Problem 40
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 4j
Problem 41
- In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector. projᵤ (v + w)
Problem 41
- In Exercises 42–43, find projᵥᵥv. Then decompose v into two vectors, v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w. v = -2i + 5j, w = 5i + 4j
Problem 42
- In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = 3i - 2j
Problem 43
- In Exercises 43–44, find the angle, in degrees, between v and w. v = 2 cos 4𝜋 i + 2 sin 4𝜋 j, w = 3 cos 3𝜋 i + 3 sin 3𝜋 j 3 3 2 2
Problem 43