2.4 The Cross Product
Chapter
Chapter 2
Section
2.4
Solutions 23 Videos

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.

1.34mins
Q1

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})

0.30mins
Q3ab

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

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

0.11mins
Q3e

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

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

0.10mins
Q3h

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

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

0.10mins
Q3k

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

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

0.10mins
Q3l

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)

Q4a

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)

0.36mins
Q4b

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)

0.35mins
Q4c

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)

0.38mins
Q4d

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

Q5

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

Q6

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}

1.05mins
Q7

Using components, show that

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

Q8a

Using components, show that

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

Q8b

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} .

2.03mins
Q9

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}

Q10a

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}

Q10b

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}

Q10d

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.

Q12

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.

5.14mins
Q13

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}.