Laws of Sines, Cosines and Vectors

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 3i + j, w = i + 3j

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

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

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j

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

In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. u ⋅ (v + w)

In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. u ⋅ v + u ⋅ w

In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. (4u) ⋅ v

In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. 4(u ⋅ v)

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree. v = 2i - j, w = 3i + 4j

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree. v = -3i + 2j, w = 4i - j

In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree. v = 6i, w = 5i + 4j

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = i + j, w = i - j

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i + 8j, w = 4i - j

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i - 2j, w = -i + j

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 3i, w = -4i

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 3i, w = -4j

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

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

If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).

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

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

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector. 5u ⋅ (3v - 4w)

In Exercises 40–41, use the dot product to determine whether v and w are orthogonal. v = 12i - 8j, w = 2i + 3j

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector. projᵤ (v + w)

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

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

In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i - 10j

In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 10j

In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 18 j 5