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.

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.

a) \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.

b) \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).

a) 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).

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

c) 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.

Introduction to Cross Product

Proof of Cross Product

ex1 Cross Product with Geometrical Vectors

ex2 Finding Cross Product of Two Algebraic Vectors

ex3 Finding Area of Triangle using Cross Product

ex4 Determining if Three Vectors are Coplanar using Cross and Dot Product

Volume of Parallelpied using Cross and Dot Product

Volume of Tetrahedron using Cross and Dot Product

V = \frac{1}{6} \vec{a} \times \vec{b} \cdot \vec{c}