7.6 The Cross Product of Two Vectors
Chapter
Chapter 7
Section
7.6
Solutions 31 Videos

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

0.53mins
Q1a

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

0.59mins
Q1b

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.

0.32mins
Q1c

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

0.26mins
Q2

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})

0.24mins
Q3a

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}

0.14mins
Q3b

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})

0.16mins
Q3c

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})

0.22mins
Q3d

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})

0.09mins
Q3e

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}

0.10mins
Q3f

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)

0.53mins
Q4a

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)

0.25mins
Q4b

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)

0.26mins
Q4c

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)

0.25mins
Q4d

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)

0.27mins
Q4e

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)

0.25mins
Q4f

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

0.28mins
Q5

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

0.22mins
Q6a

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).

0.52mins
Q6b

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

0.50mins
Q7a

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

0.47mins
Q7b

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)

1.58mins
Q8a

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)

1.21mins
Q8b

Verify each of the following:

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

Q9a

Verify each of the following:

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

Q9b

Verify each of the following:

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

1.08mins
Q9c

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?

1.50mins
Q10

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}.

1.45mins
Q11a

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.

0.17mins
Q11bc

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}).

2.46mins
Q12

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

1.19mins
Q13
Lectures on Cross Product 8 Videos

ex2 Finding Cross Product of Two Algebraic Vectors

2.08mins
ex2 Finding Cross Product of Two Algebraic Vectors

ex3 Finding Area of Triangle using Cross Product

3.27mins
ex3 Finding Area of Triangle using Cross Product

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

4.17mins
ex4 Determining if Three Vectors are Coplanar using Cross and Dot Product

Volume of Parallelpied using Cross and Dot Product

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