Let $$ \vec{a} $$ and $$ b $$ be the vectors given by $$ \vec{a}=\langle 2,0,-1 angle $$ and $$ b=\langle 1,-1,3 angle $$. (1) Compute (i) $$ \|\vec{a}\| $$ and $$ \|\vec{b}\| $$ (ii) $$ \vec{a} \cdot \vec{a}, \vec{a} \cdot \vec{b} $$ in two different ways. (iii) $$ \vec{a} imes \vec{b} $$ and $$ \vec{b} imes \vec{b} $$ (2) Assume that $$ heta $$ is the angle between the two vectors $$ \vec{a} $$ and $$ \vec{b} $$. Write down $$ \cos heta $$ in terms of $$ \vec{a} \cdot \vec{b} $$ and $$ \|\vec{a}\| $$ and $$ \|\vec{b}\| $$. Deduce $$ \cos heta $$ and find the angle $$ heta $$ between the two vectors. (3) Find an equation of the plane $$ P $$ containing $$ \vec{a} $$ and $$ \vec{b} $$

Question

Let $$ \vec{a} $$ and $$ b $$ be the vectors given by $$ \vec{a}=\langle 2,0,-1angle $$ and $$ b=\langle 1,-1,3angle $$. (1) Compute (i) $$ \|\vec{a}\| $$ and $$ \|\vec{b}\| $$ (ii) $$ \vec{a} \cdot \vec{a}, \vec{a} \cdot \vec{b} $$ in two different ways. (iii) $$ \vec{a} 	imes \vec{b} $$ and $$ \vec{b} 	imes \vec{b} $$ (2) Assume that $$ 	heta $$ is the angle between the two vectors $$ \vec{a} $$ and $$ \vec{b} $$. Write down $$ \cos 	heta $$ in terms of $$ \vec{a} \cdot \vec{b} $$ and $$ \|\vec{a}\| $$ and $$ \|\vec{b}\| $$. Deduce $$ \cos 	heta $$ and find the angle $$ 	heta $$ between the two vectors. (3) Find an equation of the plane $$ P $$ containing $$ \vec{a} $$ and $$ \vec{b} $$

Owen Davies · Accepted Answer

### (1) Compute #### (i) $$ \|\vec{a}\| $$ and $$ \|\vec{b}\| $$ $$ \|\vec{a}\| = \sqrt{5}, \|\vec{b}\| = \sqrt{11} $$ #### (ii) $$ \vec{a} \cdot \vec{a} $$ and $$ \vec{a} \cdot \vec{b} $$ $$ \vec{a} \cdot \vec{a} = 5, \vec{a} \cdot \vec{b} = -1 $$ #### (iii) $$ \vec{a} \times \vec{b} $$ and $$ \vec{b} \times \vec{b} $$ $$ \vec{a} \times \vec{b} = \langle -1, -7, -2 \rangle, \vec{b} \times \vec{b} = \vec{0} $$ ### (2) Angle $$ \theta $$ between $$ \vec{a} $$ and $$ \vec{b} $$ $$ \cos \theta = \frac{-1}{\sqrt{55}}, \theta = \cos^{-1}\left(\frac{-1}{\sqrt{55}}\right) $$ ### (3) Equation of the plane $$ P $$ containing $$ \vec{a} $$ and $$ \vec{b} $$ $$ x + 7y + 2z = 2 $$

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