Dot products and cross products

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Working with the dot product and cross product.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Draw and label two vectors $\vec {v}$ and $\vec {w}$ such that

(a): $\vec {v}\dotp \vec {w} <0$
(b): $\vec {v}\dotp \vec {w} =0$
(c): $\vec {v}\dotp \vec {w} >0$

Give three explanations (one algebraic, one geometric, one using trigonometry) why $\proj _{\vec {a}}(\vec {b}) = \vec {0}$ if $\vec {a}\dotp \vec {b} = 0$ .

Give two explanations (one algebraic and one geometric) that shows that $\vec {b} - \proj _{\vec {a}}(\vec {b}) \quad \text {is orthogonal to}\quad \vec {a}$

Let $\vec {r}$ be a random (nonzero) vector in $\R ^2$ . Compute: $\proj _\vecj \left (\proj _\veci (\vec {r})\right )$ Explain your reasoning.

Let $\vec {r}$ be a random (nonzero) vector in $\R ^2$ . Could there be a vector $\vec {v}$ such that one expects $\proj _\vecj \left (\proj _{\vec {v}}\left (\proj _\veci (\vec {r})\right )\right )\ne \vec {0}?$ Explain your reasoning.

Consider three vectors in $\R ^3$ : $\vec {a} = \vector {a_1,a_2,a_3}\quad \vec {b} = \vector {b_1,b_2,b_3}\quad \vec {c} = \vector {c_1,c_2,c_3}$ We claim: $\det \begin {bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end {bmatrix} = \vec {a}\dotp (\vec {b}\cross \vec {c}) = (\vec {a}\cross \vec {b})\dotp \vec {c}$ Give an algebraic verification of this claim. For your information, $(\vec {a}\cross \vec {b})\dotp \vec {c}$ is commonly called the scalar triple product.

Use the diagram

and recall that the volume of a parallelpiped is given by $\text {area of the base}\times \text {height}$ to answer the following:

First, explain what $\vec {a}\dotp (\vec {b}\cross \vec {c})$ and $(\vec {a}\cross \vec {b})\dotp \vec {c}$ mean geometrically and how they must be equal. There is a basic reason regarding computing volumes at play here, so you should not appeal to Problem 6 in this part).
Second, explain why it then follows that $\left |\det \begin {bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end {bmatrix}\right |$ is the volume of the parallelepiped spanned by vectors $\vec {a}$ , $\vec {b}$ , and $\vec {c}$ . You may use Problem 6 on this part.