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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`

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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`

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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.

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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.

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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})`

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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}`

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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})`

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0.16mins

Q3c

`\mathbb{R}^3`

and which are meaningless. Give a brief explanation for each.

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

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0.22mins

Q3d

`\mathbb{R}^3`

and which are meaningless. Give a brief explanation for each.

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

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0.09mins

Q3e

`\mathbb{R}^3`

and which are meaningless. Give a brief explanation for each.

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

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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)`

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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)`

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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)`

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0.26mins

Q4c

`(1, 2, 9)`

and `(-2, 3, 4)`

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0.25mins

Q4d

`(-2, 3, 3)`

and `(1, -1, 0)`

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0.27mins

Q4e

`(5, 1, 6)`

and `(-1, 2, 4)`

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0.25mins

Q4f

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

, determine `a`

.

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0.28mins

Q5

a) Calculate the vector product for `\vec{a} = (0, 1,1 )`

and `\vec{b} = (0, 5,1).`

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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)`

.

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0.52mins

Q6b

a) For the vectors `(1, 2, 1)`

and `(2, 4,2 )`

, show that their vector product is `\vec{0}`

.

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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}`

.

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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)`

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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)`

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1.21mins

Q8b

Verify each of the following:

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

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Q9a

Verify each of the following:

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

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Q9b

Verify each of the following:

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

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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?

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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}`

.

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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`

.

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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}).`

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2.46mins

Q12

Prove that `(\vec{a} - \vec{b})\times (\vec{a} + \vec{b}) = 2\vec{a} \times \vec{b}`

is true.

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1.19mins

Q13

Lectures on Cross Product
8 Videos

Introduction to Cross Product

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5.32mins

Introduction to Cross Product

Proof of Cross Product

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9.47mins

Proof of Cross Product

ex1 Cross Product with Geometrical Vectors

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1.46mins

ex1 Cross Product with Geometrical Vectors

ex2 Finding Cross Product of Two Algebraic Vectors

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2.08mins

ex2 Finding Cross Product of Two Algebraic Vectors

ex3 Finding Area of Triangle using Cross Product

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3.27mins

ex3 Finding Area of Triangle using Cross Product

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

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4.17mins

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

Volume of Parallelpied using Cross and Dot Product

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4.12mins

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}`

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3.13mins

Volume of Tetrahedron using Cross and Dot Product