Magnitude and distance:

What is the magnitude of the following vectors: $$\langle 1,-1,1 \rangle, \langle 2,-2,2 \rangle, \langle -1,0,0 \rangle?$$

What is the distince between $\langle 1,-1,1 \rangle$ and the three standard unit vectors?

Note: the standard unit vectors in three dimensions are $\langle 1,0,0 \rangle$, $\langle 0,1,0 \rangle$, and $\langle 0,0,1 \rangle$ and are also denoted by $\mathbf{i}, \mathbf{j},\mathbf{k}$.

Direction:

Challange: what is the angle between $\langle 1,-1,1 \rangle$ and the three unit vectors?

Before you do any work, which two angles will be the same?

Vector Arithmetic:

We will work with the following vectors: $$v_1 =\langle 1,3,1 \rangle, v_2 = \langle -3,-3,-3 \rangle, v_3 = \langle 1,-1,1\rangle$$

Compute the following: $$v_1+v_3, \hspace{5mm} \frac{-2}{3}v_2$$ Does the following equality hold: $$ v_1+v_3 = \frac{-2}{3} v_2?$$ Why?

More on vector equality

Are the following vectors equal? How do they fail?

$$\langle 1,-1,1 \rangle, \langle 2,-2,2 \rangle?$$

$$\langle 1,-1,1 \rangle, \langle -1,1,-1 \rangle?$$

$$\langle 1,-1,1 \rangle, \langle -1,0,0 \rangle?$$

$$\langle 1,-1,1 \rangle, \langle 1,-1,1 \rangle?$$