If vectors $v ⃗=⟨4,3⟩$ and $u ⃗=⟨9,1⟩$, calculate $v ⃗⋅u ⃗$.

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

If vectors $v⃗=12î$ and $u⃗=100ĵ$, calculate $u ⃗⋅v ⃗$.

If vectors $a⃗=13î$, $⃗b⃗=5î-12ĵ$, and $c⃗=24ĵ$, calculate $b ⃗⋅(a ⃗-c ⃗)$.

If vectors $|a⃗|=3$ and $|b⃗|=7$, and $a⃗\(\cdot\) b⃗=14.85$, determine the angle between vectors $a ⃗$ and $b ⃗$.

If vectors $a⃗=4î$ and $b⃗=3î-2ĵ$, determine the angle between vectors $a ⃗$ and $b ⃗$.