8. Vectors

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