Textbook

Calculus and Vectors Nelson
Chapter

Chapter 7
Section

Applications of Vectors Chapter Review

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Solutions
70 Videos

Given that `\vec{a}=(-1,2,1), \vec{b}=(-1,0,1),`

and `\vec{c}=(-5,4,5)`

, determine each of the following:

`\vec{a}\times\vec{b}`

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0.37mins

Q1a

Given that `\vec{a}=(-1,2,1), \vec{b}=(-1,0,1),`

and `\vec{c}=(-5,4,5)`

, determine each of the following:

`\vec{b}\times\vec{c}`

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0.33mins

Q1b

Given that `\vec{a}=(-1,2,1), \vec{b}=(-1,0,1),`

and `\vec{c}=(-5,4,5)`

, determine each of the following:

`|\vec{a}\times\vec{b}|\times|\vec{b}\times\vec{c}|`

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0.33mins

Q1c

`\vec{a}=(-1,2,1), \vec{b}=(-1,0,1),`

and `\vec{c}=(-5,4,5)`

, determine each of the following:

Why is it possible to conclude that the vectors `\vec{a},\vec{b}`

, and `\vec{c}`

are coplanar?

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0.59mins

Q1d

Given that `\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

a) `|\vec{a}|`

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Q2a

Given that `\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

`|\vec{b}|`

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Q2b

Given that `\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

`|\vec{a}-\vec{b}|`

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Q2c

`\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

`|\vec{a}+\vec{b}|`

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0.33mins

Q2d

`\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

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

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Q2e

`\vec{i},\vec{j},`

and `\vec{k}`

represent the standard basis vectors, `\vec{a}=2\vec{i}-\vec{j}+2\vec{k}`

and `\vec{b}=6\vec{i}+3\vec{j}-2\vec{k}`

, determine each of the following:

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

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1.00mins

Q2f

For what value(s) of `a`

are the vectors `\vec{x}=(3,a,9)`

and `\vec{y}=(a,12,18)`

collinear?

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2.50mins

Q3a

`\vec{x} = (3, a, 9), \vec{y} = (a, 12, 18)`

For what value(s) of `a`

are these vectors perpendicular?

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Q3b

Determine the angle between the vectors `\vec{x}=(4,5,20)`

and `\vec{y}=(-3,6,22)`

.

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1.08mins

Q4

A parallelogram has its sides determined by `\vec{OA}=(5,1)`

and `\vec{OB}=(-1,4).`

Draw a sketch of the paralleogram.

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1.19mins

Q5a

A parallelogram has its sides determined by `\vec{OA}=(5,1)`

and `\vec{OB}=(-1,4).`

Determine the angle between the two diagonals of this parallelogram.

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Q5b

An object of mass 10 kg is suspended by two pieces of rope that make an angle of 30`^\circ`

and 45`^\circ`

with the horizontal. Determine the tension in each of the two pieces of rope.

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Q6

An airplane has a speed of 300 km/h and is headed due west. A wind is blowing from the south at 50 km/h. Determine the resultant velocity of the airplane.

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1.23mins

Q7

The diagonals of a parallelogram are determined by the vectors `\vec{x}=(3,-3,5)`

and `\vec{y}=(-1,7,5)`

.

- Construct
`x,y`

and`z`

coordinates axes and draw the two given vectors. In addition, draw the parallelogram formed by these vectors.

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2.55mins

Q8a

The diagonals of a parallelogram are determined by the vectors `\vec{x}=(3,-3,5)`

and `\vec{y}=(-1,7,5)`

.

- Determine the area of the parallelogram.

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Q8b

Determine the components of a unit vector perpendicular to `(0,3,-5)`

and to `(2,3,1)`

.

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2.13mins

Q9

A triangle has vertices `A(2,3,7), B(0,-3,4),`

and `C(5,2,-4).`

- Determine the largest angle in the triangle.

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Q10a

A triangle has vertices `A(2,3,7), B(0,-3,4),`

and `C(5,2,-4).`

Determine the area of `\bigtriangleup ABC`

.

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Q10b

A mass of 10 kg is suspended by two pieces of string, 30 cm and 40 cm long, from two points that are 50 cm apart and at the same level. Find the tension in each piece of string.

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Q11

A particle is acted upon by the following four forces: 25 N pulling east, 30 N pulling west, 54 N pulling north, and 42 N pulling south.

**a)** Draw a diagram showing these four forces.

**b)** Calculate the resultant and equilibrant of these forces.

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Q12

A rectangular box is drawn as shown in the diagram below. The length of the edges of the box are `AB=4, BC=2,`

and `BF=3`

.

- Select an appropriate origin, and then determine coordinates for the other vertices.

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Q13a

A rectangular box is drawn as shown in the diagram below. The length of the edges of the box are `AB=4, BC=2,`

and `BF=3`

.

- Determine the angle between
`\vec{AF}`

and`\vec{AC}`

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1.19mins

Q13b

A rectangular box is drawn as shown in the diagram below. The length of the edges of the box are `AB=4, BC=2,`

and `BF=3`

.

- Determine the scalar projection of
`\vec{AF}`

on`\vec{AC}`

.

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Q13c

If `\vec{a}`

and `\vec{b}`

are unit vectors, and `|\vec{a}+\vec{b}|=\sqrt{3}`

, determine `(2\vec{a}-5\vec{b})\cdot(\vec{b}+3\vec{a})`

.

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Q14

Kayla wishes to swim from one side of a river, which has a current speed of 2 km/h, to a point on the other side directly opposite from her starting point. She can swim at a speed of 3 km/h in still water.

a) At what angle to the bank should Kayla swim if she wishes to swim directly across?

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Q15a

Kayla wishes to swim from one side of a river, which has a current speed of 2 km/h, to a point on the other side directly opposite from her starting point. She can swim at a speed of 3 km/h in still water.

- If the river has a width of 300 m, how long will it take for her to cross the river?

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Q15b

Kayla wishes to swim from one side of a river, which has a current speed of 2 km/h, to a point on the other side directly opposite from her starting point. She can swim at a speed of 3 km/h in still water.

- If Kayla's speed and the river's speed had been reversed, explain why it would not have been possible for her to swim across the river.

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Q15c

A parallelogram has its sides determined by the vectors `\vec{OA}=(3,2,-6)`

and `\vec{OB}=(-6,6,-2)`

.

Determine the coordinates of vectors representing the diagonals.

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Q16a

A parallelogram has its sides determined by the vectors `\vec{OA}=(3,2,-6)`

and `\vec{OB}=(-6,6,-2)`

.

- Determine the angle between the sides of the parallelogram.

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Q16b

You are given the vectors `\vec{p}=(2,-3,-3)`

, and `\vec{q}(a,b,6)`

.

- Determine values of
`a`

and`b`

if`\vec{q}`

is collinear with`\vec{p}`

.

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Q17a

You are given the vectors `\vec{p}=(2,-3,-3)`

, and `\vec{q}= (a,b,6)`

.

Determine an algebraic condition for `\vec{p}`

and `\vec{q}`

to be perpendicular.

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Q17b

You are given the vectors `\vec{p}=(2,-3,-3)`

, and `\vec{q}=(a,b,6)`

.

- Using the answer from part (b), determine the components of a unit vector that is perpendicular to
`\vec{p}`

.

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Q17c

For the vectors `\vec{m}=(\sqrt{3},-2,-3)`

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

, determine the following:

- the angle between these two vectors, to the nearest degree.

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Q18a

For the vectors `\vec{m}=(\sqrt{3},-2,-3)`

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

, determine the following:

- the scalar projection of
`\vec{n}`

on`\vec{m}`

.

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Q18b

For the vectors `\vec{m}=(\sqrt{3},-2,-3)`

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

, determine the following:

- the vector projection of
`\vec{n}`

on`\vec{m}`

.

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Q18c

For the vectors `\vec{m}=(\sqrt{3},-2,-3)`

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

, determine the following:

- the angle that
`\vec{m}`

makes with the`z`

-axis

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Q18d

A number of unit vectors, each of which is perpendicular to the other vectors in the set, is said to form a `special`

set. Determine which of the following sets are special.

**a)** `(1,0,0), (0,0,-1), (0,1,0)`

**b)** `\left( \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0 \right), \left( \frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right), (0,0,-1)`

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Q19

If `\vec{p}=\vec{i}-2\vec{j}+\vec{k}, \vec{q}=2\vec{i}-\vec{j}+\vec{k}`

and `\vec{r}=\vec{j}-2\vec{k}`

, determine each of the following:

`\vec{p}\times\vec{q}`

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Q20a

If `\vec{p}=\vec{i}-2\vec{j}+\vec{k}, \vec{q}=2\vec{i}-\vec{j}+\vec{k}`

and `\vec{r}=\vec{j}-2\vec{k}`

, determine each of the following:

`(\vec{p}-\vec{q})\times(\vec{p}+\vec{q})`

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Q20b

If `\vec{p}=\vec{i}-2\vec{j}+\vec{k}, \vec{q}=2\vec{i}-\vec{j}+\vec{k}`

and `\vec{r}=\vec{j}-2\vec{k}`

, determine each of the following:

`(\vec{p}\times\vec{r})\cdot\vec{r}`

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Q20c

`\vec{p}=\vec{i}-2\vec{j}+\vec{k}, \vec{q}=2\vec{i}-\vec{j}+\vec{k}`

and `\vec{r}=\vec{j}-2\vec{k}`

, determine each of the following:

`(\vec{p}\times\vec{q})\cdot\vec{r}`

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Q20d

Two forces of equal magnitude act on an object so that the angle between their directions is 60`^\circ`

. If their resultant has a magnitude of 20 N, find the magnitude of the equal forces.

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Q21

Determine the components of a vector that is perpendicular to the vectors `\vec{a}=(3,2,-1)`

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

.

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Q22

If `|\vec{x}|=2`

and `|\vec{y}|=5`

, determine the dot product between `\vec{x}-2\vec{y}`

and `\vec{x}+3\vec{y}`

if the angle between `\vec{x}`

and `\vec{y}`

is 60`^\circ`

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Q23

The magnitude of the scalar projection of `(1,m,0)`

on `(2,2,1)`

is 4. Determine the value of `m`

.

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Q24

Determine the angle that the vector `\vec{a}=(12,-3,4)`

makes with the `y-axis`

.

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Q25

A rectangular solid measuring `3`

by `4`

by `5`

is placed on a coordinate axis as shown in the diagram.

- Determine the coordinates of points
`C`

and`F`

.

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Q26a

A rectangular solid measuring `3`

by `4`

by `5`

is placed on a coordinate axis as shown in the diagram.

- Determine
`\vec{CF}`

. Refer to a)

`\to`

**a)** Determine the coordinates of points `C`

and `F`

.

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Q26b

A rectangular solid measuring `3`

by `4`

by `5`

is placed on a coordinate axis as shown in the diagram.

*Determine the angle between the vectors `\vec{CF}`

and `\vec{OP}`

. Refer to parts a) and b)

`\to`

**a)** Determine the coordinates of points `C`

and `F`

.

`\to`

**b)** Determine `\vec{CF}`

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Q26c

The vectors `\vec{d}`

and `\vec{e}`

are such that `|\vec{d}|=3`

and `|\vec{e}|=5`

, where the angle between the two given vectors is `50^\circ`

. Determine each of the following:

`|\vec{d}+\vec{e}|`

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Q27a

The vectors `\vec{d}`

and `\vec{e}`

are such that `|\vec{d}|=3`

and `|\vec{e}|=5`

, where the angle between the two given vectors is `50^\circ`

. Determine each of the following:

`|\vec{d}-\vec{e}|`

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Q27b

The vectors `\vec{d}`

and `\vec{e}`

are such that `|\vec{d}|=3`

and `|\vec{e}|=5`

, where the angle between the two given vectors is `50^\circ`

. Determine each of the following:

`|\vec{e}-\vec{d}|`

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Q27c

Find the scalar and vector projections of `\vec{i}+\vec{j}`

on each of the following vectors:

`\vec{i}`

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Q28a

Find the scalar and vector projections of `\vec{i}+\vec{j}`

on each of the following vectors:

`\vec{j}`

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Q28b

Determine which of the following are unit vectors:

`\displaystyle{\vec{a}=\left(\frac{1}{2},\frac{1}{3},\frac{1}{6}\right),\vec{b}=\left(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}\right),\vec{c}=\left(\frac{1}{2},\frac{-1}{\sqrt{2}},\frac{1}{2}\right)}`

, and `\displaystyle{\vec{d}=(-1,1,1)}`

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Q29a

`\displaystyle{\vec{a}=\left(\frac{1}{2},\frac{1}{3},\frac{1}{6}\right),\vec{b}=\left(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}\right),\vec{c}=\left(\frac{1}{2},\frac{-1}{\sqrt{2}},\frac{1}{2}\right)}`

, and `\displaystyle{\vec{d}=(-1,1,1)}`

Which one of vectors `\vec{a},\vec{b}`

or `\vec{c}`

is perpendicular to vector `\vec{d}`

? Explain.

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Q29b

A 25 N force is applied at the end of a 60 cm wrench. If the force makes a `30^\circ`

angle with the wrench, calculate the magnitude of the torque.

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Q30

Verify that the vectors `\vec{a}=(2,5,-1)`

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

are perpendicular.

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Q31a

Find the direction cosines for each vector. Refer to part a)

`\to`

**a)** Verify that the vectors `\vec{a}=(2,5,-1)`

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

are perpendicular.

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Q31b

If `\vec{m_1}=(\cos\alpha_a,\cos\beta_a,\cos\gamma_a)`

, the direction cosines for `\vec{a}`

, and if `\vec{m_2}=(\cos\alpha_b,\cos\beta_b,\cos\gamma_b)`

, the direction cosines for `\vec{b_2}`

, verify that `\vec{m_1}\cdot\vec{m_2}=0`

`\to`

**a)** Verify that the vectors `\vec{a}=(2,5,-1)`

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

are perpendicular.

`\to`

**b)** Find the direction cosines for each vector.

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Q31c

The diagonals of quadrilateral `ABCD`

are `3\vec{i}+3\vec{j}+10\vec{k}`

and `-\vec{i}+9\vec{j}-6\vec{k}`

. Show that quadrilateral `ABCD`

is a rectangle.

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Q32

The vector makes an angle of `30^\circ`

with the `x`

-axis and equal angles with both the `y`

-axis and `z`

-axis.

**a)** Determine the direction cosines for .

**b)** Determine the angle that makes with the `z`

-axis.

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Q33

The vectors `\vec{a}`

and `\vec{b}`

are unit vectors that make an angle of `60^\circ`

with each other. If `\vec{a}-3\vec{b}`

and `m\vec{a}+\vec{b}`

are perpendicular, determine the value of `m`

.

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Q34

If `\vec{a}=(0,4,-6)`

and `\vec{b}=-(-1,-5,-2)`

, verify that `\vec{a}\cdot\vec{b}=\frac{1}{4}|\vec{a}+\vec{b}|^2-\frac{1}{4}|\vec{a}-\vec{b}|^2`

.

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Q35

Use the fact that `|\vec{c}|^2=\vec{c}\cdot\vec{c}`

to prove the cosine law for the triangle shown in the diagram with sides `\vec{a}`

, `\vec{b}`

, and `\vec{c}`

.

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Q36

Find the lengths of the sides, the cosines of the angles, and the area of the triangle whose vertices are `A(1,-2,1)`

, `B(3,-2,5)`

, and `C(2,-2,3)`

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Q37