Textbook

Calculus and Vectors Nelson
Chapter

Chapter 6
Section

Introduction to Vectors Chapter Review

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Solutions
38 Videos

Determine whether each of the following statements is true or false. Provide a brief explanation for each answer.

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

b. `|\vec{a} + \vec{b}| = |\vec{a} + \vec{c}|`

implies` |\vec{b}| = |\vec{c}|`

c. `\vec{a} + \vec{b} = \vec{a} + \vec{c}`

implies `\vec{b} = \vec{c}`

d. `\vec{RF} = \vec{SW}`

implies `\vec{RS}= \vec{FW}`

e. `m\vec{a} + n\vec{a} = (m + n)\vec{a}`

f. If `|\vec{a}| = |\vec{b}|`

and `|\vec{c}| = |\vec{d}|`

, then `|\vec{a} + \vec{b}| =|\vec{c} + \vec{d}|`

.

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4.30mins

Q1

If `\vec{x} = 2\vec{a} - 3\vec{b} -4\vec{c}, \vec{y} = -2\vec{a} + 3\vec{b} + 3\vec{c}`

, and `\vec{z} = 2\vec{a} - 3\vec{b} + 5\vec{c}`

, determine simplified expressions for the following:

```
\displaystyle
2\vec{x} - 3\vec{y} + 5\vec{z}
```

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1.54mins

Q2a

If `\vec{x} = 2\vec{a} - 3\vec{b} -4\vec{c}, \vec{y} = -2\vec{a} + 3\vec{b} + 3\vec{c}`

, and `\vec{z} = 2\vec{a} - 3\vec{b} + 5\vec{c}`

, determine simplified expressions for the following:

```
\displaystyle
3(-2\vec{x} -4\vec{y} + \vec{z}) -(2\vec{x} -\vec{y} + \vec{z}) -2(-4\vec{x} -5\vec{y} + \vec{z})
```

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1.45mins

Q2b

If `X(-2, 1, 2)`

and `Y(-4, 4, 8)`

are two points in `\mathbb{R^3}`

, determine the following:

a. ```
\displaystyle
\vec{XY}
```

and ```
\displaystyle
|\vec{XY} |
```

b. The coordinates of a unit vector in the same direction as `\vec{XY}`

.

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3.20mins

Q3

`X(-1, 2, 6)`

and `Y(5, 5, 12)`

are two points in `\mathbb{R^3}`

a. Determine the components of a position vector equivalent to `\vec{YX}`

.

b. Determine the components of a unit vector that is in the same direction as `\vec{YX}`

.

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2.41mins

Q4

Find the components of the unit vector with the opposite direction to that of the vector from M(2, 3, 5) to N(8, 1, 2).

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2.15mins

Q5

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

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

.

a. Determine the components of the vectors representing the diagonals.

b. Determine the angles between the sides of the parallelogram.

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4.52mins

Q6

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Show that the area of triangle ABC.

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2.27mins

Q7a

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Calculate the area of triangle ABC.

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1.23mins

Q7b

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Calculate the perimeter of triangle ABC.

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1.09mins

Q7c

The points `A(-1, 1, 1)`

, `B(2, 0, 3)`

, and `C(3, 3, -4)`

are vertices of a triangle.

Calculate the coordinates of the fourth vertex D that completes the rectangle of which A, B, and C are the other three vertices.

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3.31mins

Q7d

The vectors `\vec{a}, \vec{b}`

, and `\vec{c}`

are as shown.

a. Construct the vector `\vec{a} -\vec{b} + \vec{c}`

.

b. If the vectors `\vec{a}`

and `\vec{b}`

are perpendicular, and if `|\vec{a}| = 4`

and `|\vec{b}| = 3`

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

.

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1.48mins

Q8

Given `\vec{p} = (-11, 7), \vec{q} = (-3, 1)`

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

, express each vector as a linear combination of the other two.

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2.26mins

Q9

a. Find an equation to describe the set of points equidistant from `A(2, 1, 3)`

and `B(1, 2, -3)`

.

b. Find the coordinates of two points that are equidistant from A and B.

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4.45mins

Q10

Calculate the values of `a, b`

, and `c`

in each of the following:

```
\displaystyle
2(a, b, 4) + \frac{1}{2}(6, 8, c) -3(7, c, -4) = (-24, 3, 25)
```

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2.23mins

Q11a

Calculate the values of `a, b`

, and `c`

in each of the following:

```
\displaystyle
2(a, a, \frac{1}{2}a) +(3b, 0, -5c) + 2(c, \frac{3}{2}c, 0) = (3, -22, 54)
```

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3.29mins

Q11b

Determine whether the points `A(1, -1, 1), B(2, 2, 2)`

, and `C(4, -2, 1)`

represent the vertices of a right triangle.

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1.43mins

Q12a

Determine whether the points P(1, 2, 3), Q(2, 4, 6), and R(-1, -2, -3) are collinear.

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1.25mins

Q12b

a. Show that the points A(3, 0, 4), B(1, 2, 5), and C(2, 1, 3) represent the vertices of a right triangle.

b. Determine `\cos \angle ABC`

.

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4.17mins

Q13

In the following rectangle, vectors are indicated by the direction of the arrows.

a. Name two pairs of vectors that are opposites.

b. Name two pairs of identical vectors.

c. Explain why `|\vec{AD}|^2 + |\vec{DC}|^2 = |\vec{DB}|^2`

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2.52mins

Q14

A rectangular prism measuring 3 by 4 by 5 is drawn on a coordinate axis as shown in the diagram.

a. Determine the coordinates of points C, P, E, and F.

b. Determine position vectors for `\vec{DB}`

and `\vec{CF}`

.

c. By drawing the rectangle containing `\vec{DB}`

and `\vec{OP}`

, determine the acute angle between these vectors.

d. Determine the angle between `\vec{OP}`

and `\vec{AE}`

.

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5.25mins

Q15

The vectors `\vec{a}`

and `\vec{e}`

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

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

, and the angle between them is `30^o`

. Determine each of the following:

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

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1.33mins

Q16a

The vectors `\vec{a}`

and `\vec{e}`

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

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

, and the angle between them is `30^o`

. Determine each of the following:

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

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1.17mins

Q16b

The vectors `\vec{a}`

and `\vec{e}`

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

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

, and the angle between them is `30^o`

. Determine each of the following:

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

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1.14mins

Q16c

An airplane is headed south at speed 400 km/h. The airplane encounters a wind from the east blowing at 100 km/h.

a. How far will the airplane travel in 3 h?

b. What is the direction of the airplane?

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1.55mins

Q17

Explain why the set of vectors: `\{(2, 3), (3, 5)\}`

span `\mathbb{R^2}`

.

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1.10mins

Q18a

Find `m`

and `n`

in the following: `m(2, 3) + n(3, 5) = (323, 795)`

.

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2.05mins

Q18b

Show that the vector `\vec{a} = (5, 9, 4)`

can be written as a linear combination of the vectors `\vec{b}`

and `\vec{c}`

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

and `\vec{c} = (3, 1, 4)`

. Explain why `\vec{a}`

lies in the plane determined by `\vec{b}`

and `\vec{c}`

.

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2.27mins

Q19a

Is the vector `\vec{a}= (-13, 36, 23)`

in the span of `\vec{b} = (-2, 3, 1)`

and `\vec{c} = (3, 1, 4)`

? Explain your answer.

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2.51mins

Q19b

A cube is placed so that it has three of its edges located along the positive x-, y-, and z-axes (one edge along each axis) and one of its vertices at the origin.

a. If the cube has a side length of `4`

, draw a sketch of this cube and write the coordinates of its vertices on your sketch.

b. Write the coordinates of the vector with its head at the origin and its tail at the opposite vertex.

c. Write the coordinates of a vector that starts at `(4, 4, 4)`

and is a diagonal in the plane parallel to the xz-plane.

d. What vector starts at the origin and is a diagonal in the xy-plane?

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2.16mins

Q20

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

, and `\vec{c} = 2\vec{i} + 13\vec{k}`

, determine `|2(\vec{a}+\vec{b}-\vec{c}) - (\vec{a} + 2\vec{b} ) + 3(\vec{a} - \vec{b} + \vec{c})|`

.

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2.36mins

Q21

The three points A( -3, 4), B(3, -4) and C(5, 0) are on a circle with radius 5 and centre at the origin. Points A and B are the endpoints of a diameter, and point C is on the circle.

a. Calculate `|\vec{AB}|`

, `|\vec{AC}|`

, and `|\vec{BC}|`

.

b. Show that A, B, and C are the vertices of a right triangle.

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2.41mins

Q22

In terms of `\vec{a}`

, `\vec{b}`

, `\vec{c}`

, and `\vec{0}`

, find a vector expression for each of the following:

a. `\vec{FL}`

b. `\vec{MK}`

c. `\vec{HJ}`

d. `\vec{IH} + \vec{KJ}`

e. `\vec{IK} - \vec{IH}`

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2.00mins

Q23

Draw a diagram showing the vectors `\vec{a}`

and `\vec{b}`

, where `|\vec{a}| = 2|\vec{b}|`

and `|\vec{b}| = 2|\vec{b}|`

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

are both true.

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2.08mins

Q24

If the vectors `\vec{a}`

and `\vec{b}`

are perpendicular to each other, express each of the following in terms of `|\vec{a}|`

and `|\vec{b}|`

:

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

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0.53mins

Q25a

If the vectors `\vec{a}`

and `\vec{b}`

are perpendicular to each other, express each of the following in terms of `|\vec{a}|`

and `|\vec{b}|`

:

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

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0.48mins

Q25b

If the vectors `\vec{a}`

and `\vec{b}`

are perpendicular to each other, express each of the following in terms of `|\vec{a}|`

and `|\vec{b}|`

:

`|2\vec{a}+ 3\vec{b}|`

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1.14mins

Q25c

Show that if `\vec{a}`

is perpendicular to each of the vectors `\vec{b}`

and `\vec{c}`

, then `\vec{a}`

is perpendicular to `2\vec{b} + 4\vec{c}`

.

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1.21mins

Q26