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

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

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

5u ⋅ (3v - 4w)

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 43–44, find the angle, in degrees, between v and w.

v = 2 cos(4π/3) i + 2 sin(4π/3) j, w = 3 cos(3π/2) i + 3 sin(3π/2) j

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

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 107°, C = 30°, c = 126

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1

In Exercises 22–24, sketch each vector as a position vector and find its magnitude.

v = -3j

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)

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³

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

w - v

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

||-2v||

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

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 40–41, use the dot product to determine whether v and w are orthogonal.

v = 12i - 8j, w = 2i + 3j

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