点积与正交性

Orthogonality

Introduction

The Dot Product

Orthogonal Vectors

Two nonzero vectors $x,y$ in $\mathbb{R}^n$ are orthogonal exactly when which condition holds?

What is the dot product

In one word: a vector $x$ in $\mathbb{R}^n$ whose length is $\|x\| = \sqrt{x\cdot x} = 1$ is called a ______ vector.

Compute $x\cdot y$ for $x = (1,-2,3)$ and $y = (3,1,-2)$. Enter a number.

The angle $\alpha$ between two nonzero vectors satisfies $\cos\alpha = \dfrac{x\cdot y}{\|x\|\,\|y\|}$. Find $\cos\alpha$ for $x = (1,2,2)$ and $y = (3,-2,6)$. Enter a fraction, for example 3/4.

For a vector $x$ in $\mathbb{R}^n$, the dot product $x\cdot x$ equals which quantity?

Compute the length $\|x\|$ of the vector $x = (2,-1,2)$ in $\mathbb{R}^3$. Enter a number.