Applications of Vectors Chapter Review
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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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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Q1c

Given that \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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0.25mins
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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0.23mins
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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0.41mins
Q2c

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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0.33mins
Q2d

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}\cdot\vec{b}

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0.27mins
Q2e

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}\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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0.53mins
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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1.32mins
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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5.14mins
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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3.55mins
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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2.35mins
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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1.17mins
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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5.36mins
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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1.48mins
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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1.06mins
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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0.51mins
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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1.11mins
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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0.57mins
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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0.28mins
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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1.44mins
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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1.15mins
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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0.26mins
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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0.22mins
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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1.04mins
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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0.56mins
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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0.44mins
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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0.44mins
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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0.41mins
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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0.57mins
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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0.40mins
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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0.45mins
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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1.03mins
Q20c

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})\cdot\vec{r}

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0.51mins
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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1.25mins
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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0.35mins
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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1.32mins
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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1.31mins
Q24

Determine the angle that the vector \vec{a}=(12,-3,4) makes with the y-axis.

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0.51mins
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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1.29mins
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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2.07mins
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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1.04mins
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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0.26mins
Q27c

Find the scalar and vector projections of \vec{i}+\vec{j} on each of the following vectors:

\vec{i}

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1.05mins
Q28a

Find the scalar and vector projections of \vec{i}+\vec{j} on each of the following vectors:

\vec{j}

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0.17mins
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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0.58mins
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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1.14mins
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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1.18mins
Q30

Verify that the vectors \vec{a}=(2,5,-1) and \vec{b}=(3,-1,1) are perpendicular.

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0.20mins
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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2.11mins
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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0.39mins
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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2.14mins
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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1.34mins
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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1.51mins
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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1.04mins
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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2.24mins
Q37