8. Vectors

Let $u$ be a unit vector, and let $v$ and $w$ be vectors, where $w$ is also a unit vector. Which of the following statements is true about the dot products $u\cdot v$ and $u\cdot w$?

Which of the following best defines the dot product of two vectors $a$ and $b$?

If $u$ is a unit vector, and $v$ and $w$ are also unit vectors, which of the following is always true about the dot products $u\cdot v$ and $u\cdot w$?

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

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

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 $b ⃗$.

If vectors $|v ⃗ |=12$, $|u ⃗ |=100$ and the angle between $v ⃗$ & $u ⃗$ is $\theta=\frac{\pi}{6}$, calculate $v ⃗⋅u ⃗$ .

Find the angle between each pair of vectors. Round to two decimal places as necessary.

〈2, 1〉, 〈-3, 1〉