点积

The dot product multiplies matching components and adds

The dot product measures the angle between vectors

Computing angles from components

Dot products power vector proofs

Let $\mathbf{a} = (2, -1, 3)$ and $\mathbf{b} = (1, 4, -2)$. What is the value of $\mathbf{a} \cdot \mathbf{b}$?

Two nonzero vectors satisfy $\mathbf{a} \cdot \mathbf{b} = -5$. What can you conclude about the angle $\theta$ between them?

For which value of $k$ is the vector $(3, k, -2)$ orthogonal to the vector $(k, 2, 1)$?

Let $\mathbf{a} = (2, 3, -6)$. Using $\|\mathbf{a}\|^2 = \mathbf{a} \cdot \mathbf{a}$, what is the length of $\mathbf{a}$?

For $\mathbf{a} = (1, 2, 2)$ and $\mathbf{b} = (2, -1, 2)$, give the exact value of $\cos\theta$, where $\theta$ is the angle between the vectors. Write a fraction like 4/9.

Find the value of $c$ that makes $(2, -1, 3)$ orthogonal to $(1, c, 2)$. Give just the number.