8. Vectors

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

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 $u\cdot w$?

Determine whether each pair of vectors is orthogonal.

i + 3√2j, 6i - √2j

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

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

〈1, 6〉, 〈-1, 7〉

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

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

Determine whether each pair of vectors is orthogonal.

〈1, 1〉, 〈1, -1〉

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

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

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

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