Vectors and Geometry in n Dimensions

n-tuples extend the vector toolkit to any number of dimensions

The dot product, length and angle generalize by formula

Matrices arrange vectors in a grid

Determinants package signed area and volume algebraically

Let $\mathbf{a} = (1, 2, -1, 3, 0)$ and $\mathbf{b} = (2, -1, 4, 1, 5)$ in $\mathbb{R}^5$. What is $\mathbf{a} \cdot \mathbf{b}$?

Let $\mathbf{a} = (1, -2, 2, 4)$ in $\mathbb{R}^4$. What is the length $\|\mathbf{a}\|$?

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\mathbf{x} = (1, 1)$. What is the product $A\mathbf{x}$?

Which statement about the Cauchy-Schwarz inequality holds for vectors in $\mathbb{R}^n$?

Find the value of $k$ that makes $(1, -2, 3, k)$ orthogonal to $(2, 1, -1, 1)$ in $\mathbb{R}^4$. Give just the number.

Compute the distance between the points $(1, 0, 2, 3)$ and $(4, 4, 2, 3)$ in $\mathbb{R}^4$. Give the exact value as a number or like sqrt(5).