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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.

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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})`

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0.30mins

Q3ab

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

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

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0.11mins

Q3e

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

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

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0.10mins

Q3h

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

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

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0.10mins

Q3k

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

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

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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)`

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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)`

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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)`

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0.35mins

Q4c

`(0, 2, -5)`

and `(-4, 9, 0)`

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0.38mins

Q4d

Find the unit vector perpendicular to `\vec{a} = (4, -3, 1)`

and `\vec{b}= (2, 3, -1)`

.

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Q5

Find two vectors perpendicular to both `(3, -6, 3)`

and `(-2, 4, 2)`

.

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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}`

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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}`

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Q8a

Using components, show that

`\vec{u} \times \vec{v} = \vec{0}`

if `\vec{u}`

and `\vec{v}`

are collinear

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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}
```

.

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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}`

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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}`

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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}`

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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.

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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.

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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.

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Q14

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}`

.

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Q15

Lectures
6 Videos

Introductio to Cross Product

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5.32mins

Introductio to Cross Product

Proof of Algebraic Formula for Cross Product

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9.47mins

Proof of Algebraic Formula for Cross Product

Finding Cross Product between Geometric Vectors

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1.46mins

Finding Cross Product between Geometric Vectors

Finding Cross product between Cartesian Vectors

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2.08mins

Finding Cross product between Cartesian Vectors

Finding Area of Triangle in Cartesian Coordinates

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3.27mins

Finding Area of Triangle in Cartesian Coordinates

Coplanar of Vectors

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4.17mins

Coplanar of Vectors