Answer to Question #271390 in Analytic Geometry for Wan

Question

Known vectors:

⃗a = (1, 0, 1) , ⃗b = (0, 1, -1) , ⃗c = (0, 0, 1)

Find the angle between:

1. a and b

2. a and c

3. b and c

Answer 1

$$\vec a \cdot \vec b=1(0)+0(1)+1(-1)=-1$$

$$\vec a \cdot \vec c=1(0)+0(0)+1(1)=1$$

$$\vec b \cdot \vec c=0(0)+1(0)-1(1)=-1$$

$$|\vec a|=\sqrt{1^2+0^2+1^2}=\sqrt{2}$$

$$|\vec b|=\sqrt{0^2+1^2+(-1)^2}=\sqrt{2}$$

$$|\vec c|=\sqrt{0^2+0^2+1^2}=1$$

1.

$$\cos \angle(\vec a, \vec b)=\dfrac{\vec a \cdot \vec b}{|\vec a||\vec b|}=\dfrac{-1}{\sqrt{2}(\sqrt{2})}=-\dfrac{1}{2}$$

$$\angle(\vec a, \vec b)=120\degree$$

2.

$$\cos \angle(\vec a, \vec c)=\dfrac{\vec a \cdot \vec c}{|\vec a||\vec c|}=\dfrac{1}{\sqrt{2}(1)}=\dfrac{\sqrt{2}}{2}$$

$$\angle(\vec a, \vec c)=45\degree$$

3.

$$\cos \angle(\vec b, \vec c)=\dfrac{\vec b \cdot \vec c}{|\vec b||\vec c|}=\dfrac{-1}{\sqrt{2}(1)}=-\dfrac{\sqrt{2}}{2}$$

$$\angle(\vec b, \vec c)=135\degree$$