If \vec{w} = \vec{u} \times \vec{v}, explain why \vec{w} \cdot \vec{u}, \vec{w} \cdot \vec{v}, and \vec{w} \cdot (a\vec{u} + b\vec{v}) are all zero.

State whether the following expressions are vectors, scalars, or meaningless.

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

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

State whether the following expressions are vectors, scalars, or meaningless.

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

State whether the following expressions are vectors, scalars, or meaningless.

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

State whether the following expressions are vectors, scalars, or meaningless.

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

State whether the following expressions are vectors, scalars, or meaningless.

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

Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.

(4, 0, 0) and (0, 0, 4)

Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.

(1, 2, 1) and (6, 0, 6)

Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.

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

Use the cross product to find a vector perpendicular to each of the following pairs of vectors. Check your answer using the dot product.

(0, 2, -5) and (-4, 9, 0)

Find the unit vector perpendicular to \vec{a} = (4, -3, 1) and \vec{b}= (2, 3, -1).

Find two vectors perpendicular to both (3, -6, 3) and (-2, 4, 2).

Express the unit vectors \vec{i}, \vec{j}, and \vec{k} as ordered triples and show that

a. \vec{i} \times \vec{j} = \vec{k}

b. \vec{k} \times \vec{j} = -\vec{i}

Using components, show that

\vec{u} \times \vec{v} = -\vec{v} \times \vec{u} for any vectors \vec{u} and \vec{v}

Using components, show that

\vec{u} \times \vec{v} = \vec{0} if \vec{u} and \vec{v} are collinear

Prove that \displaystyle |\vec{a} \times \vec{b}| = \sqrt{(\vec{a} \cdot \vec{a})(\vec{b}\cdot \vec{b}) - (\vec{a}\cdot \vec{b})^2} .

Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3), and \vec{c} = (4, -1, 1), calculate the following triple scalar and triple vector products.

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

Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3), and \vec{c} = (4, -1, 1), calculate the following triple scalar and triple vector products.

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

Given \vec{a} = (2, 1, 0), \vec{b} = (-1, 0, 3), and \vec{c} = (4, -1, 1), calculate the following triple scalar and triple vector products.

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

Given two non-collinear vectors \vec{a} and \vec{b}, show that \vec{a}, \vec{a} \times \vec{b}, and (\vec{a} \times \vec{b})\times \vec{a} are mutually perpendicular.

Prove that the triple scalar product of the vectors \vec{u}, \vec{v} , and \vec{w} has the property that \vec{u} \cdot (\vec{v} \times \vec{w}) = (\vec{u} \times \vec{v}) \cdot \vec{w}. Carry out the proof by expressing both sides of the equation in terms of components of the vectors.

If the cross product of \vec{a} and \vec{b} is equal to the cross product of \vec{a} and \vec{c}, this does not necessarily mean that \vec{b} equals \vec{c}. Show why this is so

a. by making an algebraic argument.

b. by drawing a geometrical diagram.

a. If \vec{a} = (1, 3, -1), \vec{b} = (2, 1, 5), \vec{v} = (-3, y, z), and \vec{a} \times \vec{v} = \vec{b}, find y and z.

a. Find another vector \vec{v} fo which \vec{a} \times \vec{v} = \vec{b}.

c. Explain why there are infinitely many vectors \vec{v} fo which \vec{a} \times \vec{v} = \vec{b}.