The two vectors \vec{a} and \vec{b} are vectors in \mathbb{R}^ 3, and \vec{a} \times \vec{b} is calculated.

Using a diagram, explain why \vec{a} \cdot (\vec{a} \times \vec{b}) = 0 and \vec{b} \cdot (\vec{a} \times \vec{b}) = 0

The two vectors \vec{a} and \vec{b} are vectors in \mathbb{R}^ 3, and \vec{a} \times \vec{b} is calculated.

Draw the parallelogram determine by \vec{a} and \vec{b}, and then draw the vector \vec{a} + \vec{b}. give a simple explanation of why (\vec{a} + \vec{b})\cdot (\vec{a}\times \vec{b}) = 0

The two vectors \vec{a} and \vec{b} are vectors in \mathbb{R}^ 3, and \vec{a} \times \vec{b} is calculated.

Why is it true that (\vec{a} + \vec{b})\cdot (\vec{a}\times \vec{b}) = 0? Explain.

For vectors in \mathbb{R}^3, explain why the calculation (\vec{a}\cdot \vec{b})(\vec{a} \times \vec{b}) = 0 is meaningless.

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

\vec{a}(\vec{b} \times \vec{c})

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

(\vec{a}\cdot \vec{b})\times \vec{c}

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

(\vec{a} \times \vec{b})\cdot (\vec{c} + \vec{d})

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

(\vec{a} \cdot \vec{b})\cdot (\vec{c} \times \vec{d})

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

(\vec{a} \times \vec{b})\times (\vec{c} \times \vec{d})

For each of the following calculations, say which are possible for vectors in \mathbb{R}^3 and which are meaningless. Give a brief explanation for each.

(\vec{a} \times \vec{b}) + \vec{c}

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(2, -3, 5) and (0, -1, 4)

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(2, -1, 3) and (3, -1, 2)

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(5, -1, 1) and (2, 4, 7)

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(1, 2, 9) and (-2, 3, 4)

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(-2, 3, 3) and (1, -1, 0)

Calculate the cross product for each of the following pairs of vectors, and verify your answer by using the dot product.

(5, 1, 6) and (-1, 2, 4)

If (-1, 3, 5) \times (0, a, 1) = (-2, 1, -1), determine a.

Calculate the vector product for \vec{a} = (0, 1,1 ) and \vec{b} = (0, 5,1).

b) Explain geometrically why it makes sense for vectors of the form (0, b ,c) and (0, d, e) to have a cross product of the form (a 0, 0).

a) For the vectors (1, 2, 1) and (2, 4,2 ), show that their vector product is \vec{0}.

b) In general, show that the vector product of two collinear vectors, (a, b, c) and (ka, kb, kc), is always \vec{0}.

In the discussion, in was states that \vec{p} \times (\vec{q} + \vec{r}) = \vec{p} \times\vec{q} + \vec{p} \times \vec{r} for vectors in \mathbb{R}^3. Verify that this rule is true for the following vectors.

\vec{p} = (1, -2, 4), \vec{q} = (1, 2, 7), and \vec{r} = (-1, 1, 0)

In the discussion, in was states that \vec{p} \times (\vec{q} + \vec{r}) = \vec{p} \times\vec{q} + \vec{p} \times \vec{r} for vectors in \mathbb{R}^3. Verify that this rule is true for the following vectors.

\vec{p} = (4, 1, 2), \vec{q} = (3, 1, -1), and \vec{r} = (0, 1, 2)

Verify each of the following:

\displaystyle \vec{i} \times \vec{j} = \vec{k} = - \vec{j} \times \vec{i}

Verify each of the following:

\displaystyle \vec{j} \times \vec{k} = \vec{i} = - \vec{k} \times \vec{j}

Verify each of the following:

\displaystyle \vec{k} \times \vec{i} = \vec{j} = - \vec{i} \times \vec{k}

Show algebraically that k(a_2b_3 - a_3b_2, a_3b_1 -a_1b_3, a_1b_2 -a_2b_1)\cdot \vec{a} = 0

What is the meaning of this result?

You are given the vectors \vec{a} = (2, 0, 0), \vec{b} = (0, 3, 0), \vec{c} = (2, 3, 0) and \vec{d} = (4, ,3, 0).

Calculate \vec{a} \times \vec{b} and \vec{c} \times \vec{d}.

You are given the vectors \vec{a} = (2, 0, 0), \vec{b} = (0, 3, 0), \vec{c} = (2, 3, 0) and \vec{d} = (4, ,3, 0).

a) Calculate (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}).

b) Without doing any calculations (that is, by visualizing the four vectors and using properties of cross products), say why (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = 0.

Show that the cross product is not associative by finding vector \vec{x}, \vec{y}, and \vec{z} such that (\vec{x} \times \vec{y}) \times \vec{z} \neq \vec{x} \times (\vec{y} \times \vec{z}).

Prove that (\vec{a} - \vec{b})\times (\vec{a} + \vec{b}) = 2\vec{a} \times \vec{b} is true.