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Solutions
27 Videos

If `\vec{a} = 3`

and `\vec{b} = 2`

, and the angle between these two vectors is `60^o`

, determine `\vec{a} \cdot \vec{b}`

.

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2.08mins

Q1

A mass of 15 kg is suspended by two cords from a ceiling. The cords have lengths of 15 cm and 20 cm, and the distance between the points where they are attached on the ceiling is 25 cm. Determine the tension in each of the two cords.

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4.59mins

Q2

In a square that has side lengths of 10 cm, what is the dot product of the vectors representing the diagonals?

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0.52mins

Q3

An airplane is travelling at 500 km/h due south when it encounters a wind from `W45^oN`

at `100 km/h`

.

a. What is the resultant velocity of the airplane?

b. How long will it take for the airplane to travel 1000 km?

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3.07mins

Q4

A 15 kg block lies on a smooth ramp that is inclined at 40° to the ground.

a. Determine the force that this block exerts in a direction perpendicular to the ramp.

b. Whatistheforce,parallel to the inclined plane,needed to prevent the block from slipping?

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2.50mins

Q5

A regular hexagon, with sides of 3 cm, is shown below. Determine `\vec{a}\cdot \vec{ b}`

.

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1.01mins

Q6

Given the vectors `\vec{a} = (4, -5, 20)`

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

, determine the following:

a. `\vec{a} \cdot \vec{b} `

b. the cosine of the angle between the two vectors

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1.24mins

Q7

Given the vectors `\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

`\vec{a} \cdot \vec{b}`

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0.31mins

Q8a

Given the vectors `\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

`\vec{b} \cdot \vec{c}`

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0.23mins

Q8b

Given the vectors `\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

`\vec{b} + \vec{c}`

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0.30mins

Q8c

`\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

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

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0.42mins

Q8d

`\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

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

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1.03mins

Q8e

`\vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}`

, and `\vec{c} = 3\vec{i} - \vec{j} - \vec{k}`

, determine the following:

`(2\vec{a} - 3\vec{b})\cdot (2\vec{a} + \vec{c})`

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1.35mins

Q8f

Given the vectors `\vec{p} = x\vec{i} + \vec{j} + 3\vec{k}`

and `\vec{q} = 3x\vec{i} + 10x\vec{j} + \vec{k}`

, determine the following:

a. the value(s) of x that make these vectors perpendicular

b. the value(s) of x that make these vectors parallel.

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2.43mins

Q9